Chapter 7 — Exercises

35 problems. Differentiation is a skill, learned by repetition. Do at least the first 25.

Part A — Power Rule (⭐)

7.1 Differentiate: (a) $x^7$ (b) $x^{1/3}$ (c) $x^{-2}$ (d) $x^{5/2}$ (e) $1$ (f) $\sqrt{x}$ (g) $1/x^3$ (h) $x^\pi$

7.2 Differentiate (sum and constant multiple): (a) $5x^3 - 2x + 7$ (b) $4 \sqrt{x} + 3/x^2$ (c) $\frac{x^4 - 3x^2 + 1}{6}$

Part B — Product and Quotient Rules (⭐⭐)

7.3 Use product rule: (a) $(x^2 + 1)(x^3 - 2)$ (b) $x^2 e^x$ (c) $x \sin x$ (d) $(2x + 3)(5 - x^2)$ (e) $\sqrt{x} \cos x$

7.4 Use quotient rule: (a) $\frac{x}{x + 1}$ (b) $\frac{x^2 + 1}{x^2 - 1}$ (c) $\frac{e^x}{x}$ (d) $\frac{\sin x}{x}$

Part C — Chain Rule (⭐⭐⭐)

7.5 Differentiate (single chain rule): (a) $(x^2 + 3)^7$ (b) $\sin(5x)$ (c) $e^{3x}$ (d) $\ln(x^2 + 1)$ (e) $\sqrt{x^2 + 1}$ (f) $\cos(\ln x)$ (g) $e^{\sin x}$ (h) $(\tan x)^4$

7.6 Differentiate (multiple chain rule): (a) $\sin(\sqrt{x^2 + 1})$ (b) $e^{\sin(x^2)}$ (c) $\ln(\cos(2x))$ (d) $(x^2 + e^x)^{10}$

Part D — Trig, Exponential, Log Combinations (⭐⭐⭐)

7.7 Differentiate: (a) $\tan(3x + 1)$ (b) $\sec(x^2)$ (c) $\arctan(2x)$ (d) $\arcsin(x/2)$ (e) $\ln(\tan x)$ (f) $2^x \cdot x^3$ (g) $(\ln x)^2$

Part E — Combined Rules (⭐⭐⭐)

7.8 Differentiate (multiple rules combined): (a) $x^2 \sin(3x)$ (b) $\frac{e^x}{\sin x}$ (c) $\sqrt{x \sin x}$ (d) $\frac{(x^2 + 1)^3}{x - 1}$ (e) $\sin^2(x) \cos(x^2)$ (f) $\frac{x e^x}{\ln(1 + x^2)}$

Part F — Logarithmic Differentiation (⭐⭐⭐⭐)

7.9 Use logarithmic differentiation: (a) $y = x^x$ (b) $y = (\sin x)^x$ (c) $y = x^{\ln x}$ (d) $y = \frac{(x+1)^3 \sqrt{x}}{(2x + 1)^2}$ (here, log makes products into sums, easier to handle)

Part G — Higher Derivatives via Rules (⭐⭐⭐)

7.10 Compute $f''(x)$: (a) $f(x) = \sin x$ — find $f''$, $f^{(3)}$, $f^{(4)}$. Find pattern. (b) $f(x) = e^x$ (c) $f(x) = x^4 - 3x^3 + x$ — find all derivatives up to $f^{(5)}$.

Part H — Tangent Lines (⭐⭐)

7.11 Find the tangent line to $y = x^3 - 2x$ at $x = 1$.

7.12 Find the tangent line to $y = e^x$ at $x = 0$.

7.13 Find the tangent line to $y = \ln x$ at $x = 1$. (This is the linear approximation of $\ln(1 + h) \approx h$ for small $h$.)

7.14 At what point on the curve $y = x^2 + x$ does the tangent line have slope $5$?

Part I — Applications (⭐⭐⭐)

7.15 A particle's position is $s(t) = t^3 e^{-t}$. Find the velocity $s'(t)$. When is the velocity zero?

7.16 A bacterial population is $P(t) = 1000 \cdot e^{0.05t}$. Find $P'(t)$. What is the population growth rate at $t = 10$ years?

7.17 A learning curve: a student's skill at $t$ hours is $S(t) = 100(1 - e^{-t/5})$. Compute $S'(t)$. What is the rate of skill improvement at $t = 5$? At $t = 10$? Does it increase or decrease over time?

7.18 Marginal cost: a firm's cost is $C(q) = 1000 + 5\sqrt{q} + 0.01 q^2$. Compute $C'(q)$.

Part J — sympy Verification (⭐⭐⭐)

7.19 Use sympy.diff to verify any 10 of the exercises above. Note where you may have made errors.

7.20 Differentiate $y = \sin(\sin(\sin x))$. Use sympy to verify your hand answer. (Three levels of chain rule.)

Part K — Speed Tests (⭐⭐⭐, time-limited)

Drill: 5 minutes total. Differentiate each in one line (no work shown):

7.21 $x^5 - 3x^2$ 7.22 $\sin(2x)$ 7.23 $e^{3x^2}$ 7.24 $x \ln x$ 7.25 $\sqrt{x + 1}$ 7.26 $\cos(x^2 + 1)$ 7.27 $\frac{1}{x^2 + 1}$ 7.28 $\tan(3x)$ 7.29 $(x^2 + 1)^5$ 7.30 $\sin^2 x$

Compare your answers with sympy. The goal is speed, not just correctness.

Part L — Reflective and Open-Ended (⭐⭐⭐⭐)

7.31 Explain in your own words why $\frac{d}{dx}(e^x) = e^x$. (The function whose rate of change equals itself.)

7.32 Why is the chain rule "the most important rule"? What sorts of derivatives can you not compute without it?

7.33 Some students learn the chain rule as "derivative of outside times derivative of inside." Others learn it as $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Which formulation works better for you? Why?

7.34 Compute $\frac{d}{dx}(\sin x)^3$ using the chain rule. Then compute $\frac{d}{dx}\sin(x^3)$. These look similar but are different. Explain.

7.35 The exponential function $e^x$ is the only function $f$ (up to constant multiples) satisfying $f' = f$. Explain why this is true. (Hint: if $g' = g$, consider $g/e^x$ and show its derivative is zero — so $g$ is a constant multiple of $e^x$.)

Total: 35 problems.

A note on Section 7.9 in the chapter (logarithmic differentiation) is the tool for Part F. Sections 7.4–7.6 (product, quotient, chain) drive Parts B–E. If a tier feels too hard, return to the matching section's worked examples before pushing on — differentiation rewards drilling more than any other topic in the book.