Calculus I · Unit 2A · review

Trigonometric, Exponential, and Logarithmic Review

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 $(1/2)^x$ .

After the explanation

Use the section idea

Reading lens

Connect each derivative formula to the function's characteristic growth, periodicity, or inverse relationship.

Mental model

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

Decision

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

Common trap

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

Check yourself

Can you distinguish a power, exponential, logarithmic, and trigonometric derivative at a glance?

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