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

The blade of a reciprocating saw has a displacement (in $\mathrm{m}$ ) of $y=2.5 \sin 8 \pi t-1.5 \cos 4 \pi t$ . Find the velocity of the blade at 2.0 s.

Find the derivative of $y=3 e^{2 x} \ln 5 x$ .

Differentiate: $y=\sqrt{\tan 2 \theta}$

Find the derivative of $y=\sqrt{\frac{e^{x}-1}{e^{x}+1}}$ .

Find $\frac{d y}{d x}$ for the implicit function: $4 \sec y=x y-2$

Find the slope of the tangent to $y=3 e^{\frac{\sin x}{4}}$ at $x=\frac{\pi}{4}$ .

Find the angle of intersection of $y=\ln x^{2}$ and $y=\ln (6-x)$ at $x=2$ .

An airplane is flying horizontally at an altitude of $5 \mathrm{~km}$ and with speed of $600 \mathrm{~km} / \mathrm{h}$ . It passes directly over an observer on the ground. Determine how fast the angle between the airplane and the observer is increasing, in rads/h, 1 minute after the airplane passed the observer.

Determine the derivative: $y = ln x^2$

Find the derivative: $y=x \cos \pi x$

Find the derivative: $y=x \arctan 2 x$

Find the maximum and minimum points of $y=e^{\sin x^{2}}$ between 0 and $\pi$ .

Find the slope of the tangent at the given value of $x: y=\log x^{2}$ at $x=5$ , to three decimal places

Find the derivative: $y=\tan 3 x \sec x$

Find the slope of the tangent to $y=2 \cos^{-1} \sqrt{x}$ at $x=0.5$ .

If $f(x)=\cos (1-\pi x)$ , find $f^{\prime \prime}\left(\frac{\pi}{2}\right)$ . Express your answer to three significant digits.

Differentiate: $y=2 \log _{b}\left(x^{2}+3\right)$

Find the absolute minimum point of $f(x)=\left(2 x^{2}-3 x-1\right) e^{-x}$ , to three decimal places.

Differentiate: $y=e^{X} \sin^{2} e^{X}$

Find $\frac{d y}{d x}$ for $\sin (x+y)=x y$ .