Purpose and Overview

Why This Chapter Matters

Learning Objectives

1. 1Recall the graphs, domains, ranges, and key properties of trigonometric, exponential, and logarithmic functions.
2. 2State and apply the derivatives of \(\sin x\), \(\cos x\), \(\tan x\), \(\sec x\), \(\csc x\), and \(\cot x\).
3. 3Apply the chain rule to differentiate compositions involving trigonometric functions.
4. 4State and apply the derivatives of \(e^x\), \(a^x\), \(\ln x\), and \(\log_a x\).
5. 5Use logarithmic differentiation to differentiate functions that would be difficult to handle otherwise.
6. 6Apply the inverse function derivative theorem to find derivatives of inverse functions.
7. 7State and apply the derivatives of the inverse trigonometric functions \(\arcsin x\), \(\arccos x\), and \(\arctan x\).
8. 8Combine all differentiation rules to differentiate complex mixed functions.
9. 9Identify and avoid common mistakes in transcendental differentiation.
10. 10Interpret derivatives of transcendental functions in modeling contexts.

Sections at a Glance

• 5.1Reviewing Special Function FamiliesTrigonometric, exponential, and logarithmic graphs; radian measure; the special role of \(e\); domain and range review
• 5.2Derivatives of \(\sin x\) and \(\cos x\)Fundamental limits; geometric interpretation; slope of sine equals height of cosine; sign of \(-\sin x\)
• 5.3Derivatives of \(\tan x\), \(\sec x\), \(\csc x\), and \(\cot x\)Quotient rule derivations; the six formulas together; co-function sign pattern
• 5.4Trigonometric Derivatives with the Chain RuleComposing trig with polynomials; vibrating strings; \(\sin^n x\) as nested composition
• 5.5Derivatives of \(e^x\)Self-referential derivative; why \(e\) is special; chain rule with exponential; bacterial growth model
• 5.6Derivatives of \(a^x\)Factor of \(\ln a\); rewriting \(a^x = e^{x \ln a}\); computational algorithms; chain rule with general base
• 5.7Derivatives of \(\ln x\)Inverse relationship derivation; slope of logarithm; chain rule gives \(g'/g\); absolute value extension
• 5.8Derivatives of \(\log_a x\)Change-of-base formula; factor \(1/(x \ln a)\); why natural log is preferred in calculus
• 5.9Logarithmic DifferentiationWhen to use it; differentiating \(x^x\); simplifying products and quotients; variable-exponent functions
• 5.10Derivatives of Inverse FunctionsInverse function derivative theorem; reciprocal slope interpretation; derivation using chain rule
• 5.11Derivatives of Inverse Trigonometric Functions\(\arcsin x\), \(\arccos x\), \(\arctan x\); derivations using Pythagorean identity; chain rule extensions; physics applications
• 5.12Mixed Transcendental DerivativesExponential times sine; log of trig; three-layer compositions; combined rules strategy
• 5.13Common Transcendental Derivative MistakesEight common errors; power rule on exponentials; missing chain rule factors; co-function signs; domain restrictions
• 5.14Preparing for Implicit and Related-Rate ProblemsPreview of implicit differentiation; related rates setup; circle example; ladder problem; chain rule as foundation

Reviewing Special Function Families

Trigonometric Functions

Exponential Functions

Logarithmic Functions

Derivatives of \(\sin x\) and \(\cos x\)

Derivatives of \(\tan x\), \(\sec x\), \(\csc x\), and \(\cot x\)

Trigonometric Derivatives with the Chain Rule

Derivatives of \(e^x\)

Derivatives of \(a^x\)

Derivatives of \(\ln x\)

Derivatives of \(\log_a x\)

Logarithmic Differentiation

Derivatives of Inverse Functions

Derivatives of Inverse Trigonometric Functions

Derivation of \(\dfrac{d}{dx}[\arcsin x]\)

Derivation of \(\dfrac{d}{dx}[\arctan x]\)

Mixed Transcendental Derivatives

Common Transcendental Derivative Mistakes

Preparing for Implicit and Related-Rate Problems

1. Differentiate \(f(x) = \ln(x^2 + y^2)\) with respect to \(x\), treating \(y\) as a constant. Now consider what would change if \(y\) were a function of \(x\).
2. The equation \(e^y = x^2 + 1\) implicitly defines \(y\) as a function of \(x\). Differentiate both sides with respect to \(x\) (chain rule on the left) and solve for \(\frac{dy}{dx}\).
3. An angle \(\theta = \arctan(x/y)\) where both \(x\) and \(y\) depend on time \(t\). Write an expression for \(\frac{d\theta}{dt}\) in terms of \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\). (You will complete this reasoning fully in Chapter 6.)

Chapter Summary

• Sec 5.1Trigonometric functions require radian measure in all derivative formulas. The number \(e\) is the unique base whose exponential function equals its own derivative. The natural log \(\ln x\) is the inverse of \(e^x\).
• Sec 5.2The derivative of \(\sin x\) is \(\cos x\); the derivative of \(\cos x\) is \(-\sin x\). The slope of the sine curve at any point equals the height of the cosine curve. Negative sign in \(-\sin x\) reflects cosine's initial decrease from its maximum.
• Sec 5.3The remaining four trig derivatives follow from the quotient rule: \(\tan' = \sec^2\), \(\sec' = \sec\tan\), \(\cot' = -\csc^2\), \(\csc' = -\csc\cot\). Co-function derivatives always carry a negative sign.
• Sec 5.4The chain rule extends every trig derivative to compositions: \(\frac{d}{dx}[\sin(g(x))] = \cos(g(x)) \cdot g'(x)\). The outer derivative and the inner derivative factor are both required.
• Sec 5.5The natural exponential function satisfies \(\frac{d}{dx}[e^x] = e^x\) — it is its own derivative. With the chain rule: \(\frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x)\). The rate of an exponential growth model at any moment equals its current value times the growth constant.
• Sec 5.6For base \(a \neq e\): \(\frac{d}{dx}[a^x] = a^x \ln a\). The factor \(\ln a\) is always present and equals 1 only when \(a = e\). This formula is derived by rewriting \(a^x = e^{x \ln a}\).
• Sec 5.7The natural log has the remarkably simple derivative \(\frac{d}{dx}[\ln x] = \frac{1}{x}\). With the chain rule, \(\frac{d}{dx}[\ln(g(x))] = g'(x)/g(x)\): the derivative of the inside over the inside.
• Sec 5.8For general base: \(\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}\). The factor \(\ln a\) in the denominator is why natural logs produce cleaner derivatives than other bases.
• Sec 5.9Logarithmic differentiation takes \(\ln\) of both sides, uses log properties to simplify, differentiates, and solves for \(y'\). It is the only correct technique for \(x^x\) and similar variable-exponent functions.
• Sec 5.10The inverse function derivative theorem: \((f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}\). Slopes become reciprocals when passing between a function and its inverse.
• Sec 5.11The three main inverse trig derivatives: \(\arcsin' = \frac{1}{\sqrt{1-x^2}}\), \(\arccos' = -\frac{1}{\sqrt{1-x^2}}\), \(\arctan' = \frac{1}{1+x^2}\). Domain restrictions apply for arcsine and arccosine. The arctangent derivative is valid for all \(x\).
• Sec 5.12Mixed functions require identifying the outermost structure and working inward. Products of transcendental functions use the product rule. Compositions of three or more layers use the chain rule repeatedly.
• Sec 5.13Eight common mistakes: power rule on exponentials; missing chain rule factors; confusing \(\ln'\) with \(e^x\); dropping \(\ln a\) factor; misapplying power or exponential rule to \(x^x\); denominator error in log derivatives; sign errors on co-function derivatives; domain violations in inverse trig.
• Sec 5.14Chapter 6 builds directly on the chain rule for transcendental functions. Implicit differentiation requires the chain rule for every implicit occurrence of \(y\). Related rates use derivatives — including transcendental ones — to connect changing quantities.

Key Terms

Trigonometric Derivative Practice

1. State the derivatives of all six trigonometric functions from memory.
2. Explain why all trigonometric derivative formulas in calculus require radian measure rather than degrees.
3. What does the graph of \(y = \cos x\) tell you about the sign of \(\frac{d}{dx}[\sin x]\) at \(x = \pi\)? Explain without computing.
4. Without computing, what is the slope of \(y = \sin x\) at its maximum? Why?
5. Explain the co-function sign pattern that connects the derivatives of cosine, cotangent, and cosecant to their counterparts.

Find each derivative.

1. \(f(x) = 4\sin x\)
2. \(g(x) = \cos x - 3x\)
3. \(h(x) = \tan x + \sec x\)
4. \(f(x) = x^2 \sin x\)
5. \(g(x) = \dfrac{\cos x}{x}\)
6. \(h(x) = \csc x - \cot x\)
7. \(f(x) = \sin(5x)\)
8. \(g(x) = \cos(x^3)\)
9. \(h(x) = \tan(2x-1)\)
10. \(f(x) = \sec(x^2)\)
11. \(g(x) = \sin^4 x\)
12. \(h(x) = \cos(\sin x)\)
13. \(f(x) = x\cos(3x)\)
14. \(g(x) = \dfrac{\sin x}{1+\cos x}\)

1. A buoy bobs up and down with height \(h(t) = 3\sin(2t)\) meters, where \(t\) is in seconds. Find the velocity at \(t = \pi/4\) seconds and interpret the result.
2. The voltage in an AC circuit is \(V(t) = 120\cos(120\pi t)\), where \(t\) is in seconds. Find the rate of change of voltage at \(t = 1/240\) seconds.
3. A pendulum has angular displacement \(\theta(t) = 0.15\cos(2\pi t)\) radians. Find both \(\theta'(t)\) and \(\theta''(t)\) and interpret each in context.

Exponential Derivative Practice

1. Explain in your own words why \(\frac{d}{dx}[e^x] = e^x\). What property of \(e\) makes this possible?
2. What is the slope of \(y = e^x\) at the point \((0,1)\)? At the point \((1, e)\)?
3. A student writes \(\frac{d}{dx}[e^{x^2}] = e^{x^2}\). What is wrong with this answer?
4. Compare \(\frac{d}{dx}[e^x]\) to \(\frac{d}{dx}[3^x]\). What is similar? What is different?

Find each derivative.

1. \(f(x) = 7e^x\)
2. \(g(x) = e^x + x^3\)
3. \(h(x) = e^{-x}\)
4. \(f(x) = e^{4x}\)
5. \(g(x) = e^{x^2+1}\)
6. \(h(x) = xe^x\)
7. \(f(x) = \dfrac{e^x}{x}\)
8. \(g(x) = e^{\sin x}\)
9. \(h(x) = e^{\cos(3x)}\)
10. \(f(x) = 2^x\)
11. \(g(x) = 5^{x^2}\)
12. \(h(x) = 3^{2x-1}\)
13. \(f(x) = x^2 e^{3x}\)
14. \(g(x) = e^{2x}\sin x\)

1. A radioactive substance decays as \(A(t) = 200e^{-0.05t}\) grams, where \(t\) is years. Find the rate of decay at \(t = 10\) and interpret the result.
2. A savings account grows as \(B(t) = 1000e^{0.06t}\) dollars. Find the instantaneous rate of growth at \(t = 5\) years.
3. A population follows \(P(t) = 300e^{0.2t}\). Compare the growth rates at \(t = 0\), \(t = 5\), and \(t = 10\). What pattern do you observe?

Logarithmic Derivative Practice

1. \(f(x) = \ln(x^4)\)
2. \(g(x) = \ln(2x+5)\)
3. \(h(x) = \ln(x^2+x)\)
4. \(f(x) = x\ln x\)
5. \(g(x) = (\ln x)^3\)
6. \(h(x) = \ln(\sin x)\)
7. \(f(x) = \ln(\sec x)\)
8. \(g(x) = \ln(e^x+1)\)
9. \(h(x) = \log_3(x^2-4)\)
10. \(f(x) = \log_{10}(5x)\)

Use logarithmic differentiation for each.

1. \(f(x) = x^x\)
2. \(g(x) = x^{\sin x}\)
3. \(h(x) = (x+1)^x\)
4. \(f(x) = \dfrac{x^2(x-1)^3}{\sqrt{x+2}}\)
5. \(g(x) = \dfrac{(x^2+1)^4 e^{2x}}{(x-3)^2}\)

1. The pH of a solution is \(\text{pH} = -\log_{10}[\text{H}^+]\). Find the rate of change of pH with respect to \([\text{H}^+]\) and interpret its sign.
2. The decibel level of a sound is \(D = 10\log_{10}(I/I_0)\). Find \(dD/dI\) and explain why the rate of change decreases as intensity increases.

Mixed Practice

Identify the outermost structure first, then apply the appropriate rules in order.

1. \(f(x) = e^x \cos x\)
2. \(g(x) = \ln(\arctan x)\)
3. \(h(x) = \sin(e^x)\)
4. \(f(x) = \arctan(\ln x)\)
5. \(g(x) = e^{\arcsin x}\)
6. \(h(x) = \dfrac{\sin x}{e^x}\)
7. \(f(x) = \ln(\sin^2 x + 1)\)
8. \(g(x) = \sec(e^{2x})\)
9. \(h(x) = \arcsin(1-x^2)\)
10. \(f(x) = x^2 \arctan(3x)\)
11. \(g(x) = e^{2x}\tan(x^2)\)
12. \(h(x) = \cos^3(\ln x)\)

Applications

1. Oscillating Spring. A spring-mass system has position \(x(t) = 0.2e^{-0.5t}\cos(4t)\) meters. Find the velocity \(x'(t)\). Describe the motion: is the amplitude increasing or decreasing? What does \(e^{-0.5t}\) contribute?
2. Learning Curve. A student's skill score follows \(S(t) = 100(1-e^{-0.1t})\) percent. Find \(S'(t)\). At what rate is the student improving after 10 days? After 50 days? What does the limiting behavior suggest?
3. Drug Concentration. The concentration \(C(t) = 5te^{-0.3t}\) mg/L where \(t\) is hours. Find \(C'(t)\) and set it to zero to find the time of maximum concentration.
4. Investment Return. An investment follows \(V(t) = Pe^{rt}\). Show that \(V'(t) = rV(t)\) and interpret: what does this equation say about the relationship between the account's level and its growth rate?
5. Projectile Angle. A ball is thrown horizontally from 20 meters. The velocity angle satisfies \(\theta = \arctan(v_y/v_x)\) where \(v_x = 15\) m/s and \(v_y = -9.8t\) m/s. Find \(d\theta/dt\) at \(t = 1\) second and interpret the result.
6. Angle of a Ramp. An engineer designs a ramp where \(\theta = \arctan(h/12)\) and \(h\) is variable. Find \(d\theta/dh\) at \(h = 3\) meters. Interpret the result.

Error Analysis

Each problem shows a student's error. Identify it, write the correct answer, and state a principle that prevents the mistake.

Chapter 5 Checkpoint