Unit 2A Practice Exam B

Derive the power rule for $x^4$ from the limit definition.

A graph of $f$ has a corner at $x=0$. Explain what must fail in the one-sided derivative limits.

Differentiate $x^3\sqrt{x^2+1}/(x-2)^4$.

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

Differentiate $\log_5(1+e^{3x})$.

Differentiate $\arctan(\sqrt{x})$.

For $x^3+y^3=6xy$, find tangent and normal lines at $(3,3)$.

If $f(2)=5$, $f'(2)=-3$, and $f"(2)=4$, derive $(f^{-1})"(5)$.

Differentiate $x^{\sin x}$ for $x>0$.

Find $f"$ for $f(x)=e^{x^2}$ and factor the result.

Find a formula for the $n$th derivative of $e^{ax}$.

Determine where the derivative of $\arcsin(2x)$ exists as a finite real number.

A calibration function is $S(V)=\ln(1+V^2)$, where $V(P)=0.5\sqrt{P+4}$. Find $dS/dP$ at $P=12$.

Explain how the chain rule becomes multiplication of local response factors in the previous problem.