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

Find the derivative of $y=5 \log \frac{x}{2}$ .

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

Find $f^{\prime}(2)$ for $f(x)=5 \cos \frac{x}{3}-x^{3}$ . Express your answer to three significant digits.

Take the logarithm of both sides and then differentiate to determine $\frac{d y}{d x}: y=(x-2)^{x}$

Find the derivative of $y=8 \sec^{-1} 2 x$ .

Find $\frac{d y}{d x}$ for $e^{x+y}-e^{x y}=x^{2}$ .

Find the second derivative: $y=\sec 2 x$

A kite rises straight up to $12.0 \mathrm{~m}$ , where it stops rising and then starts moving parallel to the ground at $2.10 \mathrm{~m} / \mathrm{s}$ . How fast is the angle between the kite string and the ground changing, in rad/s, $4.00 \mathrm{~s}$ after the kite starts to move horizontally? Assume the string is perfectly straight.

Find the derivative of $y=4^{\sqrt{x}}$ .

Find the maximum, minimum, and inflection points of the function $y=\sin x \cos x$ between $x=0$ and $x=2 \pi$ .

The energy $E$ holding together the two helium atoms of an $\mathrm{He}_{2}$ molecule is given by $E=\frac{284}{r^{6}}-\frac{10.5}{e^{2.1 r}}$ nanoJoules where $r$ is the distance between the atoms in atomic units. The atoms of one helium atom are moving apart at 1.3 atomic units per second. How fast is the energy that holds the atoms together changing when $r=4.1$ atomic units?

Find the derivative of $y=\frac{\cos x}{x^{2}}$ .

Find $\frac{d y}{d x}: e^{2 x}+x y=e^{2 y}$