Chapter 7 — Quiz

Q: B) C) D)

A) . Power rule: bring the exponent down, subtract one — . Section 7.2.

Q: B) C) D)

C) . Product rule with , : . Choice A keeps only the first term — the classic dropped-term error. Section 7.4.

Q: B) C) D)

B) . Chain rule: outer derivative times inner derivative . Choice A forgets the inner factor — the single most common chain-rule mistake. Section 7.6.

Q: for B) C) D)

A) . One of the library derivatives; valid for . Section 7.7.

Q: B) C) D)

B) . Derived in the chapter via the quotient rule from ; the Pythagorean identity collapses the numerator to . Sections 7.5 and 7.7.

Q: B) C) D)

B) . Quotient rule: . Section 7.5.

Q: B) C) D)

B) . Chain rule: outer times inner derivative . Choice A forgets the inner factor; choice D mishandles the exponent. Section 7.6.

Q: B) C) D)

C) . A constant raised to a variable is an exponential, not a power. Choice A is the tempting (wrong) power-rule answer the chapter warns against — the variable is in the exponent, so use . Sections 7.2 and 7.7.

Q: The chain rule states that equals: B) C) D)

B) . Differentiate the outer function evaluated at the inner, then multiply by the derivative of the inner. Section 7.6.

Q: is most directly computed with which rule, and equals what? Power rule only; B) Chain rule; C) Product rule; D) Quotient rule;

B) Chain rule; . Writing , the outer power gives and the inner derivative is , so . Section 7.6.

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Chapter 7 — Quiz

10 questions, ~20 minutes. Choose the single best answer, then check the worked solution. Section references point to the chapter's index.md.

1. $\dfrac{d}{dx}(x^7) = $ - A) $7x^6$ B) $7x^8$ C) $x^6$ D) $6x^7$

Answer

**A) $7x^6$.** Power rule: bring the exponent down, subtract one — $\frac{d}{dx}(x^n) = n\,x^{n-1}$. *Section 7.2.*

2. $\dfrac{d}{dx}(x^2 e^x) = $ - A) $2x e^x$ B) $x^2 e^x$ C) $(2x + x^2) e^x$ D) $2 e^x$

Answer

**C) $(2x + x^2) e^x$.** Product rule with $f = x^2$, $g = e^x$: $f'g + fg' = 2x\,e^x + x^2 e^x = (2x + x^2)e^x$. Choice A keeps only the first term — the classic dropped-term error. *Section 7.4.*

3. $\dfrac{d}{dx}(\sin(3x)) = $ - A) $\cos(3x)$ B) $3\cos(3x)$ C) $\sin(3)$ D) $-3\sin(3x)$

Answer

**B) $3\cos(3x)$.** Chain rule: outer derivative $\cos(3x)$ times inner derivative $3$. Choice A forgets the inner factor — the single most common chain-rule mistake. *Section 7.6.*

4. $\dfrac{d}{dx}(\ln x) = $ - A) $1/x$ for $x > 0$ B) $1/x^2$ C) $\ln x / x$ D) $e^x$

Answer

**A) $1/x$.** One of the library derivatives; valid for $x > 0$. *Section 7.7.*

5. $\dfrac{d}{dx}(\tan x) = $ - A) $\sec x$ B) $\sec^2 x$ C) $-\csc^2 x$ D) $\cos x / \sin^2 x$

Answer

**B) $\sec^2 x$.** Derived in the chapter via the quotient rule from $\tan x = \sin x/\cos x$; the Pythagorean identity collapses the numerator to $1$. *Sections 7.5 and 7.7.*

6. $\dfrac{d}{dx}\left(\dfrac{x}{x+1}\right) = $ - A) $1$ B) $\dfrac{1}{(x+1)^2}$ C) $\dfrac{1}{x+1}$ D) $-\dfrac{1}{x+1}$

Answer

**B) $\dfrac{1}{(x+1)^2}$.** Quotient rule: $\dfrac{(1)(x+1) - (x)(1)}{(x+1)^2} = \dfrac{1}{(x+1)^2}$. *Section 7.5.*

7. $\dfrac{d}{dx}(e^{x^2}) = $ - A) $e^{x^2}$ B) $2x\, e^{x^2}$ C) $x^2 e^{x^2}$ D) $e^{2x}$

Answer

**B) $2x\, e^{x^2}$.** Chain rule: outer $e^{x^2}$ times inner derivative $2x$. Choice A forgets the inner factor; choice D mishandles the exponent. *Section 7.6.*

8. $\dfrac{d}{dx}(2^x) = $ - A) $x\,2^{x-1}$ B) $2^x$ C) $2^x \ln 2$ D) $2^x / \ln 2$

Answer

**C) $2^x \ln 2$.** A *constant raised to a variable* is an exponential, not a power. Choice A is the tempting (wrong) power-rule answer the chapter warns against — the variable is in the exponent, so use $\frac{d}{dx}(b^x) = b^x \ln b$. *Sections 7.2 and 7.7.*

9. The chain rule states that $\dfrac{d}{dx}\,f(g(x))$ equals: - A) $f'(x) \cdot g'(x)$ B) $f'(g(x)) \cdot g'(x)$ C) $f'(g(x))$ D) $g'(f(x))$

Answer

**B) $f'(g(x)) \cdot g'(x)$.** Differentiate the outer function evaluated at the inner, then multiply by the derivative of the inner. *Section 7.6.*

10. $\dfrac{d}{dx}(\sin^2 x)$ is most directly computed with which rule, and equals what? - A) Power rule only; $2\sin x$ B) Chain rule; $2\sin x\cos x$ C) Product rule; $\cos^2 x$ D) Quotient rule; $\sec^2 x$

Answer

**B) Chain rule; $2\sin x\cos x$.** Writing $\sin^2 x = (\sin x)^2$, the outer power gives $2\sin x$ and the inner derivative is $\cos x$, so $\frac{d}{dx}(\sin x)^2 = 2\sin x\cos x = \sin(2x)$. *Section 7.6.*

Scoring

9–10: Excellent. The rules are becoming reflex — move on to Chapter 8.
7–8: Solid. Drill more chain-rule problems (Exercises Part C) before continuing.
5–6: Re-read Sections 7.4–7.6 and redo Parts B and C of the exercises.
Below 5: This is a foundational chapter. Slow down, re-read the derivations, and redo all exercise parts before continuing — everything ahead assumes this fluency.