Calculus I · 2A · review

Trigonometric, Exponential, and Logarithmic Review

Review trigonometric, exponential, and logarithmic review with mixed practice and links back to the exact lessons behind each skill.

Course progress27 of 49 core pages27 of 49

Section overview

Trigonometric, exponential, and logarithmic functions

What this section is building

Review trigonometric, exponential, and logarithmic review with mixed practice and links back to the exact lessons behind each skill.

Notice

Special-function rules preserve recognizable shapes while scaling them by a function-specific factor.

Decide

Identify the function family first, then check whether a composition requires the chain rule too.

Avoid

Using a power rule on an exponential or forgetting base and domain conditions for logarithms.

Special Functions Review

Summary

(\sin x)'=\cos x, \quad (\cos x)'=-\sin x, \quad (\tan x)'=\sec^2x,

(\cot x)'=-\csc^2x, \quad (\sec x)'=\sec x\tan x, \quad (\csc x)'=-\csc x\cot x,

(e^x)'=e^x, \quad (a^x)'=a^x\ln a, \quad (\ln|x|)'=\frac1x, \quad (\log_a x)'=\frac1{x\ln a}.

Exercise

Differentiate $x^3\cos x$ .

Exercise

Differentiate $\tan x/(x+2)$ .

Exercise

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

Exercise

Differentiate $3^x+x\ln x$ .

Exercise

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

Exercise

Find all inputs in $[0,2\pi]$ where $f(x)=\sin x+\cos x$ has a horizontal tangent.

Exercise

Explain why the derivative formulas for sine and cosine require radians.

Exercise

Compare the signs of the derivatives of $2^x$ and $(1/2)^x$ .

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.

Special Functions Review

Summary

(\sin x)'=\cos x, \quad (\cos x)'=-\sin x, \quad (\tan x)'=\sec^2x,

(\cot x)'=-\csc^2x, \quad (\sec x)'=\sec x\tan x, \quad (\csc x)'=-\csc x\cot x,

(e^x)'=e^x, \quad (a^x)'=a^x\ln a, \quad (\ln|x|)'=\frac1x, \quad (\log_a x)'=\frac1{x\ln a}.

Exercise

Differentiate $x^3\cos x$ .

Exercise

Differentiate $\tan x/(x+2)$ .

Exercise

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

Exercise

Differentiate $3^x+x\ln x$ .

Exercise

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

Exercise

Find all inputs in $[0,2\pi]$ where $f(x)=\sin x+\cos x$ has a horizontal tangent.

Exercise

Explain why the derivative formulas for sine and cosine require radians.

Exercise

Compare the signs of the derivatives of $2^x$ and $(1/2)^x$ .

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.

Special Functions Review

Related resources

Original instruction with traceable references.

Special Functions Review

Related resources

Original instruction with traceable references.