The Central Idea
A secant slope measures average change across an interval:
\[ \frac{f(x+h)-f(x)}{h}. \]If these secant slopes approach a unique value as the interval width \(h\) approaches \(0\), that limiting value is the derivative.
\[ \boxed{ f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }. \]The derivative is therefore a rule for local rate of change.
What a Derivative Represents
The derivative \(f'(x)\) can be interpreted in several equivalent ways.
- It is the local rate of change of \(f\) with respect to \(x\).
- It is the slope of the tangent line to the graph of \(f\).
- It is the limiting value of nearby secant slopes.
- It is itself a function that assigns a slope to each suitable input.
These are different views of the same mathematical quantity.
Derivative Notation
If
\[ y=f(x), \]common derivative notations include
\[ \boxed{ f'(x), \qquad y', \qquad \frac{dy}{dx} }. \]The notation \(f'(a)\) means the derivative evaluated at the particular input \(x=a\).
The Derivative Function
A derivative is often not just one number.
If
\[ f(x)=x^2, \]then different points on the parabola have different tangent slopes. The derivative function records all of those slopes:
\[ \boxed{f'(x)=2x}. \]Thus,
\[ f'(1)=2, \qquad f'(2)=4, \qquad f'(3)=6. \]First Principles
Finding a derivative directly from
\[ \lim_{h\to0}\frac{f(x+h)-f(x)}{h} \]is often called differentiating from first principles.
- Find \(f(x+h)\).
- Subtract \(f(x)\).
- Divide by \(h\).
- Simplify for \(h\ne0\).
- Determine the value approached as \(h\to0\).
Derivative of a Constant Function
Let
\[ f(x)=c, \]where \(c\) is constant. Then \(f(x+h)=c\), so
\[ \frac{f(x+h)-f(x)}{h} = \frac{c-c}{h} = 0. \]Therefore,
\[ \boxed{\frac{d}{dx}(c)=0}. \]Why a Constant Has Derivative Zero
A constant function has a horizontal graph. Its output never changes, so
\[ \Delta y=0 \]across every interval. Every secant and tangent slope is therefore
\[ \boxed{0}. \]Derivative of \(f(x)=x\)
Let \(f(x)=x\). Then
\[ f(x+h)=x+h. \]The difference quotient is
\[ \frac{(x+h)-x}{h} = \frac{h}{h} = 1. \]Therefore,
\[ \boxed{\frac{d}{dx}(x)=1}. \]Derivative of a Linear Function
Let
\[ f(x)=mx+b. \]Then
\[ f(x+h)=mx+mh+b. \]The difference quotient becomes
\[ \frac{mx+mh+b-(mx+b)}{h} = \frac{mh}{h} = m. \]Therefore,
\[ \boxed{\frac{d}{dx}(mx+b)=m}. \]Linear Functions Have Constant Derivatives
A line has the same slope everywhere. For
\[ f(x)=4x-7, \]the derivative is
\[ \boxed{f'(x)=4}. \]The derivative does not depend on \(x\) because the original function has a constant rate of change.
Derivative of \(x^2\) from First Principles
Let
\[ f(x)=x^2. \]Then
\[ f(x+h)=(x+h)^2=x^2+2xh+h^2. \]Form the difference quotient:
\[ \frac{x^2+2xh+h^2-x^2}{h} = \frac{2xh+h^2}{h} = 2x+h. \]As \(h\to0\),
\[ 2x+h\to2x. \]Therefore,
\[ \boxed{\frac{d}{dx}(x^2)=2x}. \]Interpreting \(f'(x)=2x\)
For \(f(x)=x^2\), the derivative \(f'(x)=2x\) gives the tangent slope at every input.
- At \(x=-2\), the slope is \(-4\).
- At \(x=0\), the slope is \(0\).
- At \(x=2\), the slope is \(4\).
The derivative function summarizes how the parabola changes from point to point.
Derivative of \(x^3\) from First Principles
Let
\[ f(x)=x^3. \]Expand
\[ f(x+h) = (x+h)^3 = x^3+3x^2h+3xh^2+h^3. \]The difference quotient is
\[ \frac{3x^2h+3xh^2+h^3}{h} = 3x^2+3xh+h^2. \]As \(h\to0\),
\[ 3x^2+3xh+h^2\to3x^2. \]Therefore,
\[ \boxed{\frac{d}{dx}(x^3)=3x^2}. \]A Pattern Begins to Appear
So far,
\[ \frac{d}{dx}(x)=1, \] \[ \frac{d}{dx}(x^2)=2x, \]and
\[ \frac{d}{dx}(x^3)=3x^2. \]The exponent becomes a coefficient, and the exponent on \(x\) decreases by \(1\).
The Power Rule
For powers of \(x\),
\[ f(x)=x^n, \]the derivative follows the pattern
\[ \boxed{ \frac{d}{dx}(x^n) = nx^{n-1} }. \]This is the power rule. In this lesson, we apply it primarily to the nonnegative integer powers found in polynomials.
Example: Differentiate \(x^5\)
\[ \frac{d}{dx}(x^5) = 5x^{5-1} = \boxed{5x^4}. \]Example: Differentiate \(x^8\)
\[ \boxed{ \frac{d}{dx}(x^8)=8x^7 }. \]The Constant Multiple Rule
If a function is multiplied by a constant, its derivative is multiplied by the same constant:
\[ \boxed{ \frac{d}{dx}[c\,f(x)] = c\,f'(x) }. \]For example,
\[ \frac{d}{dx}(7x^4) = 7(4x^3) = \boxed{28x^3}. \]The Sum and Difference Rules
Derivatives distribute across sums and differences:
\[ \boxed{ \frac{d}{dx}[f(x)+g(x)] = f'(x)+g'(x) } \]and
\[ \boxed{ \frac{d}{dx}[f(x)-g(x)] = f'(x)-g'(x) }. \]This allows a polynomial to be differentiated term by term.
Differentiating a Polynomial Term by Term
Consider
\[ f(x)=3x^4-5x^2+7x-9. \]Differentiate:
\[ \frac{d}{dx}(3x^4)=12x^3, \] \[ \frac{d}{dx}(-5x^2)=-10x, \] \[ \frac{d}{dx}(7x)=7, \]and
\[ \frac{d}{dx}(-9)=0. \]Therefore,
\[ \boxed{ f'(x)=12x^3-10x+7 }. \]A General Polynomial Derivative
Suppose
\[ f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0. \]Differentiating term by term gives
\[ \boxed{ f'(x) = na_nx^{n-1} +(n-1)a_{n-1}x^{n-2} +\cdots +a_1 }. \]The constant term disappears because its derivative is zero.
Differentiation Lowers Polynomial Degree
If a polynomial has degree \(n\ge1\), then its derivative has degree \(n-1\).
- A linear function differentiates to a constant.
- A quadratic differentiates to a linear function.
- A cubic differentiates to a quadratic.
- A quartic differentiates to a cubic.
This parallels the degree-lowering pattern of finite differences.
Finite Differences and Derivatives
Finite differences measure change across a nonzero step. Derivatives measure local change as the step shrinks toward zero.
For
\[ f(x)=x^3, \]a first finite difference is quadratic in \(x\), and the first derivative is
\[ f'(x)=3x^2. \]Both operations reduce polynomial degree by one, but they describe change in different ways.
Example: A Quadratic Polynomial
Let
\[ f(x)=4x^2-3x+8. \]Differentiate term by term:
\[ \boxed{ f'(x)=8x-3 }. \]Evaluating a Derivative at a Point
For
\[ f(x)=4x^2-3x+8, \]we found
\[ f'(x)=8x-3. \]At \(x=2\),
\[ f'(2)=8(2)-3=13. \]Thus, the tangent slope and local rate at \(x=2\) are
\[ \boxed{13}. \]Finding the Tangent Line from a Derivative
Suppose
\[ f(x)=x^2+1 \]and we want the tangent line at \(x=3\).
\[ f(3)=10. \]Differentiate:
\[ f'(x)=2x. \]Evaluate:
\[ f'(3)=6. \]The tangent line passes through \((3,10)\) with slope \(6\):
\[ y-10=6(x-3). \]Therefore,
\[ \boxed{y=6x-8}. \]The Derivative as a Slope Function
If
\[ f(x)=x^3-2x, \]then
\[ f'(x)=3x^2-2. \]The original function outputs heights on the graph. The derivative function outputs the corresponding local slopes.
Positive Derivatives
If
\[ f'(x)>0 \]throughout an interval, then the tangent slopes there are positive and the function is increasing on that interval.
For \(f(x)=x^2\),
\[ f'(x)=2x. \]When \(x>0\), the derivative is positive.
Negative Derivatives
If
\[ f'(x)<0 \]throughout an interval, then the tangent slopes are negative and the function is decreasing there.
For \(f(x)=x^2\), negative values of \(x\) give
\[ 2x<0. \]Zero Derivatives
If
\[ f'(a)=0, \]then the tangent line at \(x=a\) is horizontal.
For \(f(x)=x^2\),
\[ f'(x)=2x, \]so
\[ f'(0)=0. \]The parabola has a horizontal tangent at its vertex.
Derivative Units
If \(f(x)\) has output units and \(x\) has input units, then \(f'(x)\) has units
\[ \boxed{ \frac{\text{output units}}{\text{input units}} }. \]For position in meters and time in seconds, the derivative has units of meters per second.
Example: Position and Velocity
Suppose position is
\[ s(t)=2t^2+3t. \]Differentiate:
\[ s'(t)=4t+3. \]At \(t=5\),
\[ s'(5)=23. \]If position is measured in meters and time in seconds, the instantaneous velocity is
\[ \boxed{23\text{ m/s}}. \]Repeated Derivatives
A derivative is itself a function, so it may often be differentiated again.
If
\[ f(x)=x^3, \]then
\[ f'(x)=3x^2, \] \[ f''(x)=6x, \]and
\[ f'''(x)=6. \]Repeated derivatives will later help describe concavity and higher-order change.
Second-Derivative Notation
The derivative of the derivative is the second derivative. Common notations include
\[ \boxed{ f''(x), \qquad y'', \qquad \frac{d^2y}{dx^2} }. \]The main focus here remains the first derivative, but this notation prepares us for later curve analysis.
A Polynomial Eventually Differentiates to Zero
Consider
\[ f(x)=2x^3-5x^2+4x-1. \]Then
\[ f'(x)=6x^2-10x+4, \] \[ f''(x)=12x-10, \] \[ f'''(x)=12, \]and
\[ \boxed{f^{(4)}(x)=0}. \]Repeated differentiation eventually removes every polynomial term.
Derivative of a Monomial
A monomial has the form
\[ ax^n. \]Combining the constant multiple rule with the power rule gives
\[ \boxed{ \frac{d}{dx}(ax^n)=anx^{n-1} }. \]Example: Differentiate a Monomial
Differentiate
\[ f(x)=-6x^7. \]Multiply the coefficient by the exponent and reduce the exponent by one:
\[ \boxed{ f'(x)=-42x^6 }. \]Example: Differentiate a Full Polynomial
Differentiate
\[ f(x)=5x^5-4x^3+2x^2-9x+11. \]Applying the rules term by term gives
\[ \boxed{ f'(x)=25x^4-12x^2+4x-9 }. \]Differentiate First, Then Evaluate
Suppose
\[ f(x)=3x^3-2x+5 \]and we want the slope at \(x=2\).
Differentiate:
\[ f'(x)=9x^2-2. \]Then evaluate:
\[ f'(2)=9(4)-2=34. \]Therefore,
\[ \boxed{f'(2)=34}. \]Derivative Values and Function Values Are Different
For
\[ f(x)=x^2, \]the function value
\[ f(3)=9 \]gives the height of the graph at \(x=3\).
The derivative value
\[ f'(3)=6 \]gives the tangent slope there.
Height and slope answer different questions.
Common Error: Forgetting to Reduce the Exponent
The derivative of
\[ x^5 \]is not \(5x^5\). The exponent becomes the coefficient and then decreases by one:
\[ \boxed{ \frac{d}{dx}(x^5)=5x^4 }. \]Common Error: Reducing the Coefficient
For
\[ 7x^4, \]multiply the coefficient \(7\) by the exponent \(4\):
\[ 7\cdot4=28. \]The exponent, not the coefficient, decreases by \(1\):
\[ \boxed{ \frac{d}{dx}(7x^4)=28x^3 }. \]Common Error: Keeping the Constant Term
If
\[ f(x)=x^3+4, \]then
\[ \frac{d}{dx}(4)=0. \]Therefore,
\[ \boxed{f'(x)=3x^2}, \]not \(3x^2+4\).
Common Error: Treating \(x\) as a Constant
The notation
\[ \frac{d}{dx} \]means that change is being measured with respect to \(x\). Therefore,
\[ \frac{d}{dx}(x)=1, \]while a true constant such as \(7\) has derivative \(0\).
Common Error: Assuming \(f'(a)=0\) Always Means an Extremum
A zero derivative gives a horizontal tangent, but it does not automatically prove a local maximum or minimum.
For example,
\[ f(x)=x^3 \]has
\[ f'(x)=3x^2 \]and
\[ f'(0)=0. \]Yet \(x=0\) is not a local maximum or minimum.
Common Error: Using the Power Rule on a Composite Expression
For an expression such as
\[ (x^2+x)^3, \]the inside is not simply \(x\). The elementary power rule
\[ \frac{d}{dx}(x^n)=nx^{n-1} \]does not by itself justify treating the entire inner sum as though it were the variable \(x\).
More advanced differentiation rules handle composite expressions.
A Reliable Procedure for Differentiating a Simple Polynomial
- Write the polynomial in standard form if useful.
- Differentiate one term at a time.
- For \(ax^n\), multiply \(a\) by \(n\).
- Reduce the exponent by \(1\).
- Replace constant terms with \(0\).
- Preserve addition and subtraction signs.
- Combine like terms if any arise.
- Evaluate at a particular input only after finding \(f'(x)\), unless another method is required.
A Reliable Procedure for Finding a Tangent Line
- Identify the input \(x=a\).
- Compute the point \((a,f(a))\).
- Differentiate to find \(f'(x)\).
- Evaluate \(f'(a)\) to obtain the tangent slope.
- Use \[ y-f(a)=f'(a)(x-a). \]
- Simplify if desired.
- Check that the line passes through the point and has the correct slope.
Example: Full Derivative Workflow
Let
\[ f(x)=2x^4-3x^3+5x^2-7x+4. \]Differentiate each term:
\[ \frac{d}{dx}(2x^4)=8x^3, \] \[ \frac{d}{dx}(-3x^3)=-9x^2, \] \[ \frac{d}{dx}(5x^2)=10x, \] \[ \frac{d}{dx}(-7x)=-7, \]and
\[ \frac{d}{dx}(4)=0. \]Therefore,
\[ \boxed{ f'(x)=8x^3-9x^2+10x-7 }. \]Example: Full Tangent-Line Workflow
Let
\[ f(x)=x^3-2x+1 \]and find the tangent line at \(x=1\).
First,
\[ f(1)=1-2+1=0. \]Differentiate:
\[ f'(x)=3x^2-2. \]Evaluate:
\[ f'(1)=1. \]The tangent line through \((1,0)\) is
\[ y-0=1(x-1). \]Therefore,
\[ \boxed{y=x-1}. \]Practice 1: Constant
Differentiate
\[ f(x)=12. \]Practice 2: Linear Function
Differentiate
\[ f(x)=7x-4. \]Practice 3: Power Rule
Differentiate
\[ f(x)=x^6. \]Practice 4: Monomial
Differentiate
\[ f(x)=-5x^4. \]Practice 5: Polynomial
Differentiate
\[ f(x)=3x^3+2x^2-8x+6. \]Practice 6: Evaluate a Derivative
Let
\[ f(x)=x^2+4x. \]Find \(f'(3)\).
Practice 7: Tangent Line
Let
\[ f(x)=x^2-1. \]Find the tangent line at \(x=2\).
Practice 8: Position and Velocity
An object's position is
\[ s(t)=3t^2-2t+5. \]Find its instantaneous velocity at \(t=4\).
Practice 9: Repeated Derivatives
Let
\[ f(x)=x^4. \]Find \(f'(x)\), \(f''(x)\), \(f'''(x)\), and \(f^{(4)}(x)\).
Practice 10: Identify the Error
A student differentiates
\[ f(x)=4x^3+5x-8 \]and writes
\[ f'(x)=12x^3+5-8. \]Identify and correct the errors.
Practice Answers
Practice 1
\[ \boxed{f'(x)=0}. \]Practice 2
\[ \boxed{f'(x)=7}. \]Practice 3
\[ \boxed{f'(x)=6x^5}. \]Practice 4
\[ \boxed{f'(x)=-20x^3}. \]Practice 5
\[ \boxed{ f'(x)=9x^2+4x-8 }. \]Practice 6
\[ f'(x)=2x+4. \] \[ f'(3)=10. \]Therefore,
\[ \boxed{f'(3)=10}. \]Practice 7
\[ f(2)=3, \qquad f'(x)=2x, \qquad f'(2)=4. \]Hence,
\[ y-3=4(x-2), \]so
\[ \boxed{y=4x-5}. \]Practice 8
\[ s'(t)=6t-2. \] \[ s'(4)=22. \]Therefore, the instantaneous velocity is
\[ \boxed{22} \]in position units per time unit.
Practice 9
\[ f'(x)=4x^3, \] \[ f''(x)=12x^2, \] \[ f'''(x)=24x, \]and
\[ \boxed{f^{(4)}(x)=24}. \]Practice 10
The derivative of \(4x^3\) is \(12x^2\), not \(12x^3\). The derivative of the constant \(-8\) is \(0\).
Therefore,
\[ \boxed{ f'(x)=12x^2+5 }. \]What to Remember
The derivative is the local rate of change:
\[ \boxed{ f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }. \]For simple polynomials:
- constants differentiate to \(0\);
- \(\dfrac{d}{dx}(x)=1\);
- \(\dfrac{d}{dx}(x^n)=nx^{n-1}\);
- constant coefficients remain as multipliers;
- sums and differences may be differentiated term by term;
- polynomial differentiation lowers degree by one;
- \(f'(a)\) gives the tangent slope and local rate at \(x=a\);
- the sign of the derivative describes increasing or decreasing behavior;
- a zero derivative gives a horizontal tangent but does not automatically prove an extremum.
Where We Go Next
Derivatives turn local slope into a function that can be studied across an entire curve. Once the derivative is known, we can ask where the slope is positive, negative, or zero and use those patterns to locate important features of a graph.
The next lesson, Slope Functions and Critical Points, uses derivative functions to identify horizontal tangents, critical numbers, and changes in local behavior.