Multivariable Sensitivity | Malone Math | Malone Global University

The Central Idea

Suppose an output depends on several inputs:

\[ y=f(x_1,x_2,\ldots,x_n). \]

A sensitivity question asks:

How much does the output change when one or more inputs change?

In multivariable analysis, the answer depends on which inputs change, how they change together, and where in the input space the system currently is.

Local Sensitivity

The simplest sensitivity information is local.

For

\[ f(x,y), \]

the partial derivative

\[ \boxed{ f_x } \]

measures local sensitivity to \(x\) while holding \(y\) fixed.

Likewise,

\[ \boxed{ f_y } \]

measures local sensitivity to \(y\).

Coordinate Sensitivity

At a point \((a,b)\),

\[ f_x(a,b) \]

tells how much the output changes per small unit increase in \(x\), with \(y\) fixed.

Similarly,

\[ f_y(a,b) \]

measures the corresponding first-order response to \(y\).

Example: Cost Sensitivity

Suppose

\[ C(q,w) \]

gives cost as a function of production quantity \(q\) and wage rate \(w\).

Then

\[ C_q \]

measures local cost sensitivity to quantity, while

\[ C_w \]

measures local cost sensitivity to wage rate.

Sensitivity Is Point-Dependent

In nonlinear models, sensitivity can vary from place to place.

A variable may have a weak effect near one operating point and a strong effect near another.

Thus the statement

\[ f_x=10 \]

is meaningful only in the context of where it is evaluated if \(f_x\) is not constant.

Example: Nonlinear Sensitivity

Let

\[ f(x,y)=x^2y. \]

Then

\[ f_x=2xy \]

and

\[ f_y=x^2. \]

The sensitivity to \(x\) depends on both \(x\) and \(y\), while the sensitivity to \(y\) depends on \(x\).

The Gradient as a Sensitivity Vector

For a scalar-valued function,

\[ f(x,y), \]

the gradient collects the coordinate sensitivities:

\[ \boxed{ \nabla f = \langle f_x,f_y\rangle }. \]

It is the natural local sensitivity vector.

What the Gradient Adds

The gradient does more than list partial derivatives.

It also identifies:

the direction of greatest local increase;
the direction of greatest local decrease;
the maximum first-order directional sensitivity;
normal directions to level sets.

Maximum Local Sensitivity

For unit directions \(\mathbf{u}\),

\[ D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u}. \]

The largest possible value is

\[ \boxed{ \|\nabla f\| }. \]

Thus gradient magnitude measures the maximum local output change per unit movement in input space.

Direction of Greatest Sensitivity

\[ \nabla f\neq\mathbf{0}, \]

then the unit direction of greatest increase is

\[ \boxed{ \frac{\nabla f}{\|\nabla f\|} }. \]

The opposite direction gives the greatest local decrease.

Directional Sensitivity

Sometimes several inputs change together.

If the coordinated change follows the unit direction

\[ \mathbf{u}, \]

then the directional sensitivity is

\[ \boxed{ D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u} }. \]

Example: Coordinated Input Change

Suppose

\[ \nabla f(a,b)=\langle6,8\rangle. \]

In the unit direction

\[ \mathbf{u} = \left\langle \frac35,\frac45 \right\rangle, \]

the directional sensitivity is

\[ D_{\mathbf{u}}f = 6\left(\frac35\right) + 8\left(\frac45\right). \]

Therefore,

\[ \boxed{ D_{\mathbf{u}}f=10 }. \]

First-Order Combined Change

If both \(x\) and \(y\) change slightly, then

\[ \boxed{ \Delta f \approx f_x\,\Delta x + f_y\,\Delta y }. \]

This is the basic first-order sensitivity formula for simultaneous input changes.

Differential Form

The total differential is

\[ \boxed{ df = f_x\,dx + f_y\,dy }. \]

It predicts the local output response to small changes in the inputs.

Gradient Form of the Differential

We can write

\[ \boxed{ df = \nabla f \cdot d\mathbf{x} } \]

where

\[ d\mathbf{x} = \langle dx,dy\rangle. \]

This expresses local sensitivity as a dot product between model sensitivity and input change.

Example: First-Order Output Change

Let

\[ f(x,y)=x^2y. \]

\[ (2,3), \]

we have

\[ f_x=2xy=12 \]

and

\[ f_y=x^2=4. \]

\[ dx=0.01, \qquad dy=-0.02, \]

then

\[ df = 12(0.01)+4(-0.02) = 0.04. \]

\[ \boxed{ \Delta f\approx0.04 }. \]

Absolute Sensitivity

A derivative such as

\[ f_x \]

is an absolute sensitivity.

It measures output change per unit change in the input.

Its value depends on the units used for both the output and the input.

Relative Sensitivity

Sometimes we care about percentage-type changes rather than absolute changes.

For a positive output \(f\) and positive input \(x\), a normalized sensitivity can be written

\[ \boxed{ S_x = \frac{x}{f} \frac{\partial f}{\partial x} }. \]

Elasticity

The dimensionless quantity

\[ \boxed{ E_x = \frac{\partial f}{\partial x} \frac{x}{f} } \]

is often called an elasticity or normalized sensitivity.

It approximates the percentage change in \(f\) resulting from a one-percent change in \(x\), with other inputs held fixed.

Example: Elasticity of a Power Model

Let

\[ f(x)=Ax^p. \]

Then

\[ f'(x)=Ap x^{p-1}. \]

The elasticity is

\[ \frac{x}{f}f'(x) = \frac{x}{Ax^p} Ap x^{p-1}. \]

Therefore,

\[ \boxed{ E_x=p }. \]

Multivariable Power Models

Consider

\[ f(x,y)=Ax^\alpha y^\beta. \]

Then the coordinate elasticities are

\[ \boxed{ E_x=\alpha } \]

and

\[ \boxed{ E_y=\beta }. \]

The exponents directly encode normalized sensitivity.

Why Normalization Helps

Suppose one input is measured in dollars and another in kilograms.

Their raw partial derivatives may not be directly comparable.

Dimensionless normalized sensitivities can help compare relative influence across inputs with different scales and units.

Scaling Matters

Sensitivity values depend on the coordinate system.

If an input is rescaled from meters to centimeters, its numerical derivative changes.

Therefore, statements such as “input \(x\) is more sensitive than input \(y\)” require attention to units and normalization.

Local Sensitivity versus Global Influence

Partial derivatives describe local behavior.

They do not automatically tell us what happens over large changes in the inputs.

A variable with small local sensitivity may still have a large global effect if it varies over a large range.

Finite-Change Sensitivity

For a non-small change, we may compare

\[ \Delta f = f(\mathbf{x}+\Delta\mathbf{x}) - f(\mathbf{x}). \]

This finite change captures nonlinear effects that a first-order derivative may miss.

First-Order Approximation Error

The approximation

\[ \Delta f \approx \nabla f\cdot\Delta\mathbf{x} \]

is usually best when

\[ \|\Delta\mathbf{x}\| \]

is small and the function is smooth near the operating point.

Second-Order Effects

When first-order sensitivity is not enough, second derivatives describe how the sensitivities themselves change.

In two variables, this information is organized in the Hessian matrix:

\[ \boxed{ H_f = \begin{bmatrix} f_{xx}&f_{xy}\\ f_{yx}&f_{yy} \end{bmatrix} }. \]

Meaning of the Hessian

The Hessian describes local curvature and interaction in sensitivity.

For example:

\(f_{xx}\) tells how sensitivity to \(x\) changes with \(x\);
\(f_{yy}\) tells how sensitivity to \(y\) changes with \(y\);
\(f_{xy}\) measures how the \(x\)-sensitivity changes as \(y\) changes.

Interaction Sensitivity

Suppose

\[ f_{xy}\neq0. \]

Then the effect of changing \(x\) depends on the value of \(y\).

This is a local interaction between inputs.

Example: Interaction

Let

\[ f(x,y)=xy. \]

Then

\[ f_x=y \]

and

\[ f_{xy}=1. \]

The sensitivity to \(x\) increases one-for-one as \(y\) increases.

Several Outputs at Once

Many systems produce more than one output.

Suppose

\[ \mathbf{F}(x_1,\ldots,x_n) = \langle F_1,\ldots,F_m \rangle. \]

Then one gradient is no longer enough.

We need the Jacobian matrix.

Jacobian Matrix as a Sensitivity Map

The Jacobian is

\[ \boxed{ J_{\mathbf{F}} = \begin{bmatrix} \frac{\partial F_1}{\partial x_1} & \cdots & \frac{\partial F_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial F_m}{\partial x_1} & \cdots & \frac{\partial F_m}{\partial x_n} \end{bmatrix} }. \]

Each entry measures the local sensitivity of one output to one input.

Linearized Multivariable Response

For a vector-valued output,

\[ \mathbf{F}:\mathbb{R}^n\to\mathbb{R}^m, \]

small input changes satisfy

\[ \boxed{ \Delta\mathbf{F} \approx J_{\mathbf{F}} \Delta\mathbf{x} }. \]

The Jacobian acts as the local linear sensitivity operator.

Example: Two Inputs and Two Outputs

Let

\[ \mathbf{F}(x,y) = \langle x^2+y,\, xy \rangle. \]

Then

\[ J_{\mathbf{F}} = \begin{bmatrix} 2x&1\\ y&x \end{bmatrix}. \]

Evaluate the Jacobian

\[ (x,y)=(2,3), \]

we have

\[ \boxed{ J_{\mathbf{F}}(2,3) = \begin{bmatrix} 4&1\\ 3&2 \end{bmatrix} }. \]

Predicting the Output Change

Suppose

\[ \Delta\mathbf{x} = \begin{bmatrix} 0.01\\ -0.02 \end{bmatrix}. \]

Then

\[ \Delta\mathbf{F} \approx \begin{bmatrix} 4&1\\ 3&2 \end{bmatrix} \begin{bmatrix} 0.01\\ -0.02 \end{bmatrix}. \]

Therefore,

\[ \boxed{ \Delta\mathbf{F} \approx \begin{bmatrix} 0.02\\ -0.01 \end{bmatrix} }. \]

Parameter Sensitivity

A parameter is often a fixed input within one model run but a variable quantity across scenarios.

\[ y=f(x;\theta), \]

then

\[ \boxed{ \frac{\partial f}{\partial\theta} } \]

measures local sensitivity of the output to the parameter \(\theta\).

Parameters versus State Variables

A state variable may evolve within the model.

A parameter may be held fixed while solving the model.

Sensitivity analysis can still vary either one to ask how the output or solution changes.

Scenario Sensitivity

A practical sensitivity study may evaluate the model under several input scenarios:

baseline;
lower-input scenario;
higher-input scenario;
combined-input scenario;
stress scenario.

Derivatives describe local sensitivity, while scenario comparison can show larger-scale nonlinear response.

One-at-a-Time Sensitivity

A simple numerical method changes one input at a time while holding the others fixed.

This approximates the logic of partial derivatives.

It is easy to interpret but can miss interactions among inputs.

Simultaneous Sensitivity

When several inputs change together, the combined local effect is better represented by

\[ \nabla f\cdot\Delta\mathbf{x} \]

or, for several outputs,

\[ J_{\mathbf{F}}\Delta\mathbf{x}. \]

Uncertainty Propagation

Suppose measured inputs have small uncertainties.

The differential provides a first-order approximation of how those uncertainties affect the output.

For

\[ f(x,y), \]

we use

\[ \boxed{ df = f_x\,dx + f_y\,dy }. \]

Worst-Case First-Order Bound

\[ |dx|\le\delta_x, \qquad |dy|\le\delta_y, \]

then

\[ |df| \le |f_x|\delta_x + |f_y|\delta_y. \]

Thus a practical bound is

\[ \boxed{ |\Delta f| \approx |df| \lesssim |f_x|\delta_x + |f_y|\delta_y }. \]

Example: Measurement Uncertainty

Let the area of a rectangle be

\[ A=xy. \]

Then

\[ dA=y\,dx+x\,dy. \]

\[ x=10, \qquad y=5, \]

with

\[ |dx|\le0.02, \qquad |dy|\le0.03, \]

then

\[ |dA| \lesssim 5(0.02)+10(0.03) = 0.4. \]

Therefore,

\[ \boxed{ |\Delta A|\lesssim0.4 } \]

square units to first order.

Independent Random Uncertainties

If small input errors are independent random quantities with standard deviations \(\sigma_x\) and \(\sigma_y\), a common first-order approximation is

\[ \boxed{ \sigma_f^2 \approx f_x^2\sigma_x^2 + f_y^2\sigma_y^2 }. \]

This is a statistical uncertainty-propagation approximation rather than a worst-case bound.

Correlated Uncertainties

If input errors are correlated, covariance terms appear.

For two inputs,

\[ \boxed{ \operatorname{Var}(f) \approx f_x^2\operatorname{Var}(x) + f_y^2\operatorname{Var}(y) + 2f_xf_y\operatorname{Cov}(x,y) }. \]

Dependence between inputs can therefore increase or reduce output uncertainty.

Matrix Form of Uncertainty Propagation

For a scalar output with gradient \(\nabla f\) and input covariance matrix \(\Sigma\),

\[ \boxed{ \operatorname{Var}(f) \approx \nabla f^{T} \Sigma \nabla f }. \]

This is the matrix version of first-order uncertainty propagation.

Several Outputs and Covariance

\[ \mathbf{F} \]

has Jacobian \(J\), then a first-order covariance approximation is

\[ \boxed{ \Sigma_{\mathbf{F}} \approx J \Sigma_{\mathbf{x}} J^T }. \]

This formula describes how input uncertainty spreads into several correlated outputs.

Implicit Sensitivity

Sensitivity can also arise when variables are linked by an implicit relationship

\[ F(x,y)=0. \]

Then

\[ \boxed{ \frac{dy}{dx} = -\frac{F_x}{F_y} }. \]

This measures how \(y\) must respond locally to changes in \(x\) while remaining on the constraint.

Constrained Sensitivity

Under a constraint, not every input direction is feasible.

Sensitivity should then be measured only along directions tangent to the feasible set.

This distinguishes free sensitivity from constrained sensitivity.

Lagrange Multipliers and Sensitivity

In constrained optimization, a Lagrange multiplier can often be interpreted as the local sensitivity of the optimized objective value to the constraint level.

\[ g(x,y)=c, \]

then, under appropriate regularity conditions, \(\lambda\) is related to how the optimal value changes when \(c\) changes slightly.

Sensitivity Along a Path

Suppose a system moves along

\[ \mathbf{r}(t). \]

Then the rate of change of a scalar output

\[ f(\mathbf{r}(t)) \]

\[ \boxed{ \frac{d}{dt} f(\mathbf{r}(t)) = \nabla f(\mathbf{r}(t)) \cdot \mathbf{r}'(t) }. \]

Path Sensitivity Interpretation

The gradient tells how sensitive the field is in space.

The velocity tells how the system moves through that space.

Their dot product gives the actual rate experienced along the path.

Dimensionless Inputs

Sensitivity comparisons often improve when inputs are nondimensionalized.

\[ \hat{x}=\frac{x}{x_*}, \]

then \(\hat{x}\) measures \(x\) relative to a characteristic scale \(x_*\).

This can make sensitivity magnitudes easier to compare across variables.

Characteristic-Scale Sensitivity

If \(x_*\) and \(f_*\) are characteristic scales, then a scaled sensitivity may be written

\[ \boxed{ S_x^* = \frac{x_*}{f_*} \frac{\partial f}{\partial x} }. \]

This quantity is dimensionless when the units are chosen consistently.

Sensitivity Ranking

A modeler may wish to rank inputs by influence.

Possible measures include:

absolute partial derivative magnitude;
normalized sensitivity;
elasticity;
finite scenario change;
variance contribution;
maximum directional effect.

The appropriate measure depends on the question being asked.

Local Ranking Can Change

Suppose

\[ |f_x|>|f_y| \]

at one point.

At another point, the inequality may reverse.

Nonlinear models can therefore have sensitivity rankings that depend on the operating regime.

Threshold Behavior

Some models remain relatively insensitive until an input approaches a threshold, after which sensitivity rises sharply.

Large derivatives near thresholds can signal fragile or rapidly changing behavior.

Conditioning

A problem is called well-conditioned when small input changes produce small output changes relative to the relevant scales.

It is ill-conditioned when small input perturbations can produce large output changes.

Condition Numbers

A condition number is a normalized measure of sensitivity.

In simple scalar settings, a relative condition number may resemble

\[ \boxed{ \kappa = \left| \frac{x}{f(x)} f'(x) \right| }. \]

Multivariable condition measures often use vector and matrix norms.

Model Sensitivity versus Numerical Sensitivity

A model can be sensitive because of its mathematics.

A numerical algorithm can also be sensitive because of rounding, approximation, or instability.

These are related but distinct issues.

Sensitivity and Identifiability

If an output barely changes when a parameter changes, data may contain little information about that parameter.

Strong local sensitivity can make a parameter easier to detect, while weak sensitivity can make estimation difficult.

Sensitivity and Optimization

Optimization algorithms often use sensitivity information.

For example,

\[ -\nabla f \]

gives the direction of steepest local decrease.

Sensitivity therefore directly guides gradient-based optimization.

Sensitivity and Design

A design may be intentionally placed in a region where important outputs are less sensitive to uncertain inputs.

This idea is often called robust design.

Lower sensitivity can make performance less vulnerable to noise, manufacturing variation, or environmental change.

Sensitivity and Control

In a controlled system, some inputs may be adjustable while others are disturbances.

Sensitivity helps distinguish:

which controls have strong effect;
which disturbances matter most;
which outputs are most fragile;
which directions produce the largest response.

Sensitivity and Model Reduction

If a model output is consistently insensitive to certain inputs over the range of interest, those inputs may sometimes be fixed, simplified, or removed in an approximate model.

Such simplification requires domain-specific judgment and validation.

Local versus Global Sensitivity Methods

Local methods use derivatives near one operating point.

Global methods explore broader ranges of inputs.

Local methods are often faster and analytically clearer, while global methods are better at revealing nonlinear interactions and regime changes.

Finite-Difference Sensitivity

If derivatives are unavailable, a partial derivative can be approximated by

\[ \boxed{ f_x(x,y) \approx \frac{ f(x+h,y)-f(x,y) }{h} }. \]

Central differences often improve accuracy:

\[ \boxed{ f_x(x,y) \approx \frac{ f(x+h,y)-f(x-h,y) }{2h} }. \]

Choosing the Step Size

If \(h\) is too large, the finite difference may not reflect local behavior.

If \(h\) is too small in floating-point computation, roundoff error may become significant.

Numerical sensitivity requires balancing these effects.

Automatic and Symbolic Differentiation

In computational work, sensitivities may be obtained through:

symbolic differentiation;
automatic differentiation;
finite differences;
adjoint methods for large systems.

The underlying derivative concepts remain the same even when software performs the calculations.

Common Error: Comparing Raw Derivatives with Different Units

A derivative measured in dollars per kilogram is not directly comparable to one measured in dollars per hour without considering scale and context.

Normalize when necessary before ranking sensitivities.

Common Error: Treating Local Sensitivity as Global Truth

A partial derivative describes behavior near the evaluation point.

It should not automatically be extrapolated across a large region of the input space.

Common Error: Ignoring Interactions

One-at-a-time partial derivatives can miss how variables reinforce or offset one another.

Mixed partials, directional derivatives, scenario studies, and global methods can reveal these interactions.

Common Error: Ignoring the Sign

The magnitude of a derivative measures strength of local response.

The sign tells direction:

positive means increasing input locally increases output;
negative means increasing input locally decreases output.

Common Error: Ignoring the Operating Point

In a nonlinear model,

\[ f_x(a,b) \]

may be very different from

\[ f_x(c,d). \]

Always state or track the point at which sensitivity is evaluated.

Common Error: Confusing Uncertainty with Sensitivity

Sensitivity measures how strongly an output responds.

Uncertainty measures how uncertain the inputs or outputs are.

A highly sensitive model may still have precise outputs if the inputs are known very accurately.

A weakly sensitive model may still have uncertain outputs if the input uncertainty is large.

Common Error: Ignoring Feasible Directions

Under constraints, some input changes are impossible.

Sensitivity should then be interpreted along feasible tangent directions, not arbitrary directions in the full input space.

A Reliable Local Sensitivity Procedure

Identify the output and all relevant inputs.
Choose the operating point.
Compute the partial derivatives.
Assemble the gradient or Jacobian if needed.
Evaluate the derivatives at the operating point.
Check signs, magnitudes, units, and scales.
Use differentials or matrix multiplication for simultaneous changes.
Normalize when comparing inputs with different scales.
Check whether nonlinear or interaction effects require broader analysis.

A Reliable Uncertainty Procedure

Identify uncertain inputs.
Estimate their magnitudes, standard deviations, or covariance structure.
Compute the local gradient or Jacobian.
Use a worst-case differential bound or covariance propagation formula as appropriate.
Check whether the perturbations are small enough for a first-order approximation.
Compare the linearized estimate with direct scenario evaluation when possible.

Example: Full Scalar Sensitivity Workflow

Let

\[ P(x,y)=5x^2y+2y^2. \]

The partial derivatives are

\[ P_x=10xy \]

and

\[ P_y=5x^2+4y. \]

\[ (x,y)=(2,3), \]

we have

\[ P_x=60 \]

and

\[ P_y=32. \]

Interpret the Full Scalar Workflow

Near \((2,3)\), holding the other input fixed:

a one-unit increase in \(x\) changes \(P\) at a local rate of about \(60\);
a one-unit increase in \(y\) changes \(P\) at a local rate of about \(32\).

The gradient is

\[ \boxed{ \nabla P(2,3) = \langle60,32\rangle }. \]

Combined Change in the Full Scalar Workflow

\[ dx=0.01, \qquad dy=-0.02, \]

then

\[ dP = 60(0.01)+32(-0.02). \]

Therefore,

\[ \boxed{ dP=-0.04 }. \]

The two input effects nearly cancel.

Example: Full Vector Sensitivity Workflow

Let

\[ \mathbf{F}(x,y) = \langle x^2+y,\, xy+y^2 \rangle. \]

Its Jacobian is

\[ J_{\mathbf{F}} = \begin{bmatrix} 2x&1\\ y&x+2y \end{bmatrix}. \]

Evaluate the Full Vector Workflow

\[ (1,2), \]

we obtain

\[ \boxed{ J_{\mathbf{F}}(1,2) = \begin{bmatrix} 2&1\\ 2&5 \end{bmatrix} }. \]

Predict a Vector Output Change

Suppose

\[ \Delta\mathbf{x} = \begin{bmatrix} 0.02\\ -0.01 \end{bmatrix}. \]

Then

\[ \Delta\mathbf{F} \approx \begin{bmatrix} 2&1\\ 2&5 \end{bmatrix} \begin{bmatrix} 0.02\\ -0.01 \end{bmatrix}. \]

Therefore,

\[ \boxed{ \Delta\mathbf{F} \approx \begin{bmatrix} 0.03\\ -0.01 \end{bmatrix} }. \]

Practice 1: Coordinate Sensitivity

Let

\[ f(x,y)=x^2y+3y. \]

Find \(f_x\) and \(f_y\).

Practice 2: Evaluate Sensitivity

For

\[ f(x,y)=x^2y, \]

evaluate \(f_x\) and \(f_y\) at \((2,3)\).

Practice 3: Gradient Magnitude

\[ \nabla f(a,b)=\langle3,4\rangle, \]

find the maximum directional sensitivity.

Practice 4: First-Order Change

Suppose

\[ f_x(a,b)=5, \qquad f_y(a,b)=-2. \]

Estimate \(\Delta f\) when

\[ \Delta x=0.03, \qquad \Delta y=0.01. \]

Practice 5: Elasticity

Let

\[ f(x)=7x^3. \]

Find the elasticity

\[ E_x = \frac{x}{f}f'(x). \]

Practice 6: Jacobian Sensitivity Matrix

Let

\[ \mathbf{F}(x,y) = \langle x+y,\, xy \rangle. \]

Find \(J_{\mathbf{F}}\).

Practice 7: Vector Output Change

\[ J = \begin{bmatrix} 2&1\\ -1&3 \end{bmatrix} \]

and

\[ \Delta\mathbf{x} = \begin{bmatrix} 0.1\\ 0.2 \end{bmatrix}, \]

estimate \(\Delta\mathbf{F}\).

Practice 8: Worst-Case Error Bound

Suppose

\[ f_x=4, \qquad f_y=-3 \]

and

\[ |dx|\le0.02, \qquad |dy|\le0.01. \]

Find a first-order worst-case bound for \(|\Delta f|\).

Practice 9: Implicit Sensitivity

\[ F(x,y)=x^2+y^2-25=0, \]

find \(dy/dx\).

Practice 10: Identify the Error

A student says:

“Because \(|f_x|>|f_y|\), the output is always more sensitive to \(x\) than to \(y\).”

Explain why this conclusion may be incomplete.

Practice Answers

Practice 1

\[ \boxed{ f_x=2xy } \]

and

\[ \boxed{ f_y=x^2+3 }. \]

Practice 2

We have

\[ f_x=2xy \]

and

\[ f_y=x^2. \]

At \((2,3)\),

\[ \boxed{ f_x=12, \qquad f_y=4 }. \]

Practice 3

The maximum directional sensitivity is the gradient magnitude:

\[ \sqrt{3^2+4^2}=5. \]

Therefore,

\[ \boxed{5}. \]

Practice 4

\[ \Delta f \approx 5(0.03) + (-2)(0.01). \]

Therefore,

\[ \boxed{ \Delta f\approx0.13 }. \]

Practice 5

Since

\[ f'(x)=21x^2, \]

we have

\[ E_x = \frac{x}{7x^3}(21x^2) = 3. \]

Therefore,

\[ \boxed{E_x=3}. \]

Practice 6

\[ \boxed{ J_{\mathbf{F}} = \begin{bmatrix} 1&1\\ y&x \end{bmatrix} }. \]

Practice 7

\[ \Delta\mathbf{F} \approx \begin{bmatrix} 2&1\\ -1&3 \end{bmatrix} \begin{bmatrix} 0.1\\ 0.2 \end{bmatrix}. \]

Therefore,

\[ \boxed{ \Delta\mathbf{F} \approx \begin{bmatrix} 0.4\\ 0.5 \end{bmatrix} }. \]

Practice 8

\[ |\Delta f| \lesssim 4(0.02)+3(0.01). \]

Therefore,

\[ \boxed{ |\Delta f|\lesssim0.11 }. \]

Practice 9

We have

\[ F_x=2x, \qquad F_y=2y. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = -\frac{x}{y} } \]

wherever \(y\neq0\).

Practice 10

The conclusion may fail because:

the derivatives may have different units;
the input variables may operate on different scales;
the comparison may hold only at one point;
normalized sensitivity may reverse the ranking;
large finite changes may reveal nonlinear effects not visible locally.

What to Remember

Multivariable sensitivity is the study of local and broader model response to changes in several inputs.

Partial derivatives measure coordinate sensitivities.
The gradient collects scalar-output sensitivities into one vector.
Directional derivatives measure response along coordinated input directions.
Differentials estimate combined first-order output changes.
Elasticities and normalized sensitivities help compare variables with different scales.
Mixed partials reveal interaction in local sensitivity.
Jacobians organize sensitivity when there are several outputs.
First-order derivatives support uncertainty propagation and local condition analysis.
Local sensitivity should not be confused with global influence.
Units, scales, constraints, and the operating point all matter.

The central synthesis is

\[ \boxed{ \text{input change} \xrightarrow{\text{gradient or Jacobian}} \text{predicted output change} }. \]

Where We Go Next

This lesson completes the Multivariable Relationships sequence by bringing together the central derivative-based tools for functions of several variables.

From here, the same ideas support broader work in modeling, optimization, numerical analysis, dynamical systems, probability, engineering, economics, and scientific computation.

The key habit is to ask not only what a model predicts, but also how that prediction changes when the inputs, parameters, or constraints change.