Calculus I · Unit 2A · practice

Unit 2A Cumulative Derivative Practice

Practice method

Work in three passes

First, classify. Name the derivative idea or rule before writing algebra. This separates a recognition error from a calculation error.

Second, solve without the key. Record a complete attempt, including bounds, constants, domains, units, or interpretation when the prompt asks for them.

Third, reveal one answer at a time. Compare the first line where your work differs, close the answer, and redo that item from a blank start.

Cumulative Practice Set

Part A: Meaning and first principles

Exercise

Explain the difference between $f(3)$ and $f'(3)$ .

Answer reveal

Exercise 1 answer

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Exercise

Use the limit definition to find $f'(2)$ for $f(x)=x^2+1$ .

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Exercise 2 answer

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Exercise

Use the limit definition to find $g'(1)$ for $g(x)=1/x$ .

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Exercise 3 answer

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Exercise

Find the tangent and normal lines to $y=x^2-2x$ at $x=3$ .

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Exercise 4 answer

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Exercise

If $C'(100)=7.5$ for a cost function in dollars, interpret the value with units.

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Exercise 5 answer

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Exercise

Sketch a continuous function that is not differentiable at one point and explain the failure.

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Exercise 6 answer

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Part B: Powers and algebraic rules

Exercise

Differentiate $7x^8-3x^{1/2}+5x^{-4}$ .

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Exercise 7 answer

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Exercise

Differentiate $(x^2+1)(x^3-2x)$ .

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Exercise 8 answer

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Exercise

Differentiate $(3x-1)/(x^2+4)$ .

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Exercise 9 answer

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Exercise

Differentiate $x^3\sqrt{x}+4/x^2$ after rewriting with exponents.

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Exercise 10 answer

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Exercise

Find the tangent line to $x^4-2x^2$ at $x=1$ .

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Exercise 11 answer

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Exercise

Find every input where $f(x)=x^3-3x$ has a horizontal tangent.

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Exercise 12 answer

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Part C: Trigonometric, exponential, and logarithmic functions

Exercise

Differentiate $x^2\sin x$ .

Answer reveal

Exercise 13 answer

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Exercise

Differentiate $\tan x/(x+1)$ .

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Exercise 14 answer

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Exercise

Differentiate $(x^2-1)e^x$ .

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Exercise 15 answer

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Exercise

Differentiate $5^x+\log_3x$ .

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Exercise 16 answer

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Exercise

Find the tangent line to $y=\ln x$ at $x=e$ .

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Exercise 17 answer

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Exercise

Find all $x\in[0,2\pi]$ where $(\sin x+\cos x)'=0$ .

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Exercise 18 answer

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Part D: Chain rule and combined rules

Exercise

Differentiate $(5x^2-1)^7$ .

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Exercise 19 answer

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Exercise

Differentiate $\sqrt{1+x^4}$ .

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Exercise 20 answer

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Exercise

Differentiate $\cos(3x^2)$ .

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Exercise 21 answer

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Exercise

Differentiate $e^{\sin x}$ .

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Exercise 22 answer

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Exercise

Differentiate $\ln(x^3-4x)$ .

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Exercise 23 answer

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Exercise

Differentiate $x^3\tan(2x)$ .

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Exercise 24 answer

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Exercise

Differentiate $\sin(x^2)/(x^2+1)$ .

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Exercise 25 answer

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Exercise

Differentiate $1/(1+e^{-x})$ .

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Exercise 26 answer

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Part E: Implicit, inverse, logarithmic, and higher derivatives

Exercise

Find $y'$ for $x^3+y^3=6xy$ .

Answer reveal

Exercise 27 answer

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Exercise

Find the tangent line to $x^2+xy+y^2=7$ at $(1,2)$ .

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Exercise 28 answer

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Exercise

Find $y"$ for $x^2+y^2=9$ .

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Exercise 29 answer

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Exercise

If $f(3)=8$ and $f'(3)=-2$ , find $(f^{-1})'(8)$ .

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Exercise 30 answer

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Exercise

Differentiate $\arcsin(2x)$ .

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Exercise 31 answer

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Exercise

Differentiate $\arctan(x^2)$ .

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Exercise 32 answer

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Exercise

Use logarithmic differentiation on $x^2(x+1)^3/(x^2+4)$ .

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Exercise 33 answer

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Exercise

Differentiate $(x+2)^{x^2}$ .

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Exercise 34 answer

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Exercise

Find $f"$ for $f(x)=x^2e^x$ .

Answer reveal

Exercise 35 answer

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Exercise

Find $g^{(3)}$ for $g(x)=x\cos x$ .

Answer reveal

Exercise 36 answer

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After the explanation

Use the section idea

Reading lens

Mixed practice tests recognition: the page title no longer tells you which rule to use.

Mental model

A dependable solution separates interpretation, rule selection, calculation, and verification.

Decision

Classify first, calculate second, and use answer reveals to diagnose the first incorrect decision.

Common trap

Reading the solution before making a complete attempt or treating every miss as mere algebra.

Check yourself

Can you explain why your method fits before comparing your final answer?

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