Calculus I · 2A · practice

Unit 2A Cumulative Derivative Practice

Practice unit 2a cumulative derivative practice with mixed problems, staged guidance, and source-linked solutions.

Course progressSupporting resourceReturn to the unit map →

Section overview

Review, practice, exams, and reference

What this section is building

Practice unit 2a cumulative derivative practice with mixed problems, staged guidance, and source-linked solutions.

Notice

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

Decide

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

Avoid

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

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 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

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 2 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 3 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 4 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 5 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 6 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Part B: Powers and algebraic rules

Exercise

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

Answer reveal

Exercise 7 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 8 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 9 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 10 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 11 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 12 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Part C: Trigonometric, exponential, and logarithmic functions

Exercise

Differentiate $x^2\sin x$ .

Answer reveal

Exercise 13 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 14 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 15 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $5^x+\log_3x$ .

Answer reveal

Exercise 16 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 17 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 18 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Part D: Chain rule and combined rules

Exercise

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

Answer reveal

Exercise 19 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 20 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 21 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $e^{\sin x}$ .

Answer reveal

Exercise 22 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 23 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 24 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 25 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 26 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

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

Exercise

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

Answer reveal

Exercise 27 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 28 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 29 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 30 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $\arcsin(2x)$ .

Answer reveal

Exercise 31 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $\arctan(x^2)$ .

Answer reveal

Exercise 32 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 33 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 34 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 35 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 36 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Calculus I · 2A · practice

Unit 2A Cumulative Derivative Practice

Practice unit 2a cumulative derivative practice with mixed problems, staged guidance, and source-linked solutions.

Course progressSupporting resourceReturn to the unit map →

Section overview

Review, practice, exams, and reference

What this section is building

Practice unit 2a cumulative derivative practice with mixed problems, staged guidance, and source-linked solutions.

Notice

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

Decide

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

Avoid

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

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 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

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 2 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 3 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 4 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 5 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 6 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Part B: Powers and algebraic rules

Exercise

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

Answer reveal

Exercise 7 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 8 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 9 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 10 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 11 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 12 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Part C: Trigonometric, exponential, and logarithmic functions

Exercise

Differentiate $x^2\sin x$ .

Answer reveal

Exercise 13 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 14 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 15 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $5^x+\log_3x$ .

Answer reveal

Exercise 16 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 17 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 18 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Part D: Chain rule and combined rules

Exercise

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

Answer reveal

Exercise 19 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 20 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 21 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $e^{\sin x}$ .

Answer reveal

Exercise 22 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 23 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 24 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 25 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 26 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

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

Exercise

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

Answer reveal

Exercise 27 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 28 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 29 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 30 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $\arcsin(2x)$ .

Answer reveal

Exercise 31 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $\arctan(x^2)$ .

Answer reveal

Exercise 32 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 33 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 34 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 35 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

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

Answer reveal

Exercise 36 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.