Exam 33: Derivatives of Trigonometric, Logarithmic, and Exponential Functions

Find the second derivative of the function $y=\sin x \cos x$ .

Determine the derivative: $y=\ln \cos x$

Find the second derivative of $y=e^{-x^{2}}$ .

Differentiate: $y = tan^{-1}\sqrt{1+x^2}$

Find $\frac{d y}{d x}$ of $3 x^{2}-2 \csc (y-x)=1$ .

Find the second derivative of $y=\frac{e^{4 x^{2}}}{2}$ .

Differentiate implicitly: $x y^{2}=x \cos y+y \sin x$

Differentiate: $y=\csc^{n} a x$

Find the derivative of $y=5 x^{4}-3 \cos 6 x$ .

Find the value of the derivative of $y=4 \tan x^{2} at x=\frac{1}{4}$ .

Find the derivative: $y=\sqrt{x} \operatorname{arccsc} x$

Differentiate: $y=\sqrt{e^{x}+e^{-x}}$

The potential energy of a spring with a constant of $k$ , a maximum amplitude of $A$ , and an angular velocity of $\omega$ is given by $P=\frac{1}{2} k A^{2} \cos^{2} \omega t$ (joules) where $t$ is in seconds. Find the rate of change in the potential energy of a mass on a spring with a constant of $1.2 \mathrm{~N} / \mathrm{m}$ , a maximum amplitude of $0.53 \mathrm{~m}$ , and an angular velocity of $0.031 \mathrm{rad} / \mathrm{s}$ , when $t=5.8 \mathrm{~s}$ .

Take the second derivative of the function $y=\cos e^{x}$ .

Find the equation of the tangent to the curve $y=\frac{2}{e^{x}}$ at $x=0$ .

Differentiate: $y=5 x \tan^{-1} \frac{1}{x}$

Find the derivative: $y=\sin^{3} x \cos x$

Find $\frac{d y}{d x}$ for the implicit function: $x^{2}+y^{2}=\sin x y$

Differentiate: $y=\ln \sqrt{x^{2}+4 x+1}$

Find the slope of the tangent to the curve $y=x-\arcsin x^{2}$ at $x=0.5$ .