Exam 3: Applications of Differentiation

Differentiate the function. $h(t)=\frac{\ln 6 t}{\ln 12 t}$

A 20-ft ladder leaning against a wall begins to slide. How fast is the angle between the ladder and the wall changing at the instant of time when the bottom of the ladder is 13 ft from the wall and sliding away from the wall at the rate of 4 ft/sec? Round the answer to the nearest hundredth. (figure not to scale)

Let $f(x)=x^{2}-4 x+4$ . a. Find the derivative $f^{\prime}$ of b. Find the point on the graph of f where the tangent line to the curve is horizontal. c. Sketch the graph of f and the tangent line to the curve at the point found in part (b). d. What is the rate of change of f at this point?

Find the derivative of the function. $f(x)=\sin ^{-1}\left(e^{8 x}\right)$

Differentiate the function. $g(\theta)=\ln |\tan 5 \theta|$

Find the derivative of the function. $f(x)=\left(5 x+6 e^{x}\right)\left(x-e^{x}\right)$

Find an equation of the tangent line to the curve $y-\ln \left(x^{7}+y^{7}\right)=0$ at $(1,0)$ .

A water trough is 20 m long and a cross-section has the shape of an isosceles trapezoid that is 20 cm wide at the bottom, 60 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of $0.7 \mathrm{~m}^{3} / \min$ , how fast is the water level rising when the water is 25 cm deep? Round the result to the nearest hundredth.

Differentiate. $K(x)=\left(3 x^{6}+1\right)\left(x^{11}-16 x\right)$

A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm. The area is A(x). Find $A^{\prime}$ ( 15 ).

Find an equation of the tangent line to the curve $y=9 \tan x$ at the point $\left(\frac{\pi}{4}, 9\right)$ .

Differentiate. $y=\frac{\sin x}{3+\cos x}$

Suppose that f and g are functions that are differentiable at x = -3 and that f (-3) = 3, $f^{\prime}$ (-3) = -5, g (-3) = 3, and $g^{\prime}$ (-3) = 3. Find $h^{\prime}(-3)$ . $h(x)=\frac{x f(x)}{x+g(x)}$

Differentiate the function. $G(u)=\ln \sqrt{\frac{9 u+81}{9 u-81}}$

If $h(2)=7$ and $h^{\prime}(2)=-5$ , find $\left.\frac{d}{d x}\left(\frac{h(x)}{x}\right)\right|_{x=2}$

Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0018$ cm thick to a hemispherical dome with diameter $50$ m.

Differentiate the function. $y=x^{8} \log _{8} \sqrt[8]{x^{8}-9}$

Find $f^{\prime}$ in terms of $g^{\prime}$ . $f(x)=x^{5} g(x)$

Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is $F=\frac{G m M}{r^{2}}$ . Find $\frac{d F}{d r}(7)$ .

Find the given derivative by finding the first few derivatives and observing the pattern that occurs. $\frac{d^{97}}{d x^{97}}(\sin x)$