Chapter 3

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ G(x)=\frac{2 x-1}{x-4} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{3}{x^{3}}+x^{-4} $$

A certain bacterial culture is growing so that it has a mass of \(\frac{1}{2} t^{2}+1\) grams after \(t\) hours. (a) How much did it grow during the interval \(2 \leq t \leq 2.01 ?\) (b) What was its average growth rate during the interval \(2 \leq t \leq 2.01 ?\) (c) What was its instantaneous growth rate at \(t=2 ?\)

Find \(D_{x} y\). $$ y=\operatorname{coth}^{-1}\left(x^{5}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \ln \sqrt{3 x-2} $$

The vertex angle \(\theta\) opposite the base of an isosceles triangle with equal sides of length 100 centimeters is increasing at \(\frac{1}{10}\) radian per minute. How fast is the area of the triangle increasing when the vertex angle measures \(\pi / 6\) radians? Hint: \(A=\frac{1}{2} a b \sin \theta\)

Find a formula for $$ D_{x}^{n}\left(a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\right) $$

Find \(D_{x} y\). $$ y=\left(2-3 x^{2}\right)^{4}\left(x^{7}+3\right)^{3} $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\sec ^{3} x $$

Find \(D_{x} y\) using the rules of this section. $$ y=2 x^{-6}+x^{-1} $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ G(x)=\frac{2 x}{x^{2}-x} $$

A business is prospering in such a way that its total (accumulated) profit after \(t\) years is \(1000 t^{2}\) dollars. (a) How much did the business make during the third year (between \(t=2\) and \(t=3) ?\) (b) What was its average rate of profit during the first half of the third year, between \(t=2\) and \(t=2.5 ?\) (The rate will be in dollars per year.) (c) What was its instantaneous rate of profit at \(t=2 ?\)

Find \(D_{x} y\). $$ y=x \cosh ^{-1}(3 x) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ \frac{d y}{d x} \text { if } y=3 \ln x $$

Find \(d y / d x\). \(y=3 x^{5 / 3}+\sqrt{x}\)

Without doing any calculating, find each derivative. (a) \(D_{x}^{4}\left(3 x^{3}+2 x-19\right)\) (b) \(\quad D_{x}^{12}\left(100 x^{11}-79 x^{10}\right)\) (c) \(D_{x}^{11}\left(x^{2}-3\right)^{5}\)

Find \(D_{x} y\). $$ y=\frac{(x+1)^{2}}{3 x-4} $$

$$ \begin{array}{l} \text { C } \text { . Find the equation of the tangent line to } y=\cos x \text { at }\\\ x=1 \end{array} $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ g(x)=\sqrt{3 x} $$

Find \(D_{x} y\). $$ y=x^{2} \sinh ^{-1}\left(x^{5}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ \frac{d y}{d x} \text { if } y=x^{2} \ln x $$

Water is pumped at a uniform rate of 2 liters \((1\) liter \(=1000\) cubic centimeters \()\) per minute into a tank shaped like a frustum of a right circular cone. The tank has altitude 80 centimeters and lower and upper radii of 20 and 40 centimeters, respectively (Figure 11 ). How fast is the water level rising when the depth of the water is 30 centimeters? Note: The volume, \(V\), of a frustum of a right circular cone of altitude \(h\) and lower and upper radii \(a\) and \(b\) is \(V=\frac{1}{3} \pi h \cdot\left(a^{2}+a b+b^{2}\right)\).

Find \(d y / d x\). \(y=\sqrt[3]{x}-2 x^{7 / 2}\)

Find a formula for \(D_{x}^{n}(1 / x)\).

Find \(D_{x} y\). $$ y=\frac{2 x-3}{\left(x^{2}+4\right)^{2}} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{3}{x^{3}}-\frac{1}{x^{4}} $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ g(x)=\frac{1}{\sqrt{3 x}} $$

Suppose that the revenue \(R(n)\) in dollars from producing \(n\) computers is given by \(R(n)=0.4 n-0.001 n^{2} .\) Find the instantaneous rates of change of revenue when \(n=10\) and \(n=100\). (The instantaneous rate of change of revenue with respect to the amount of product produced is called the marginal revenue.)

Find \(D_{x} y\). $$ y=\ln \left(\cosh ^{-1} x\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ \frac{d z}{d x} \text { if } z=x^{2} \ln x^{2}+(\ln x)^{3} $$

Water is leaking out the bottom of a hemispherical tank of radius 8 feet at a rate of 2 cubic feet per hour. The tank was full at a certain time. How fast is the water level changing when its height \(h\) is 3 feet? Note: The volume of a segment of height \(h\) in a hemisphere of radius \(r\) is \(\pi h^{2}[r-(h / 3)] .\)

Find \(d y / d x\). \(y=\sqrt[3]{x}+\frac{1}{\sqrt[3]{x}}\)

If \(f(x)=x^{3}+3 x^{2}-45 x-6\), find the value of \(f^{\prime \prime}\) at each zero of \(f^{\prime}\), that is, at each point \(c\) where \(f^{\prime}(c)=0\).

Find the indicated derivative. \(y^{\prime}\) where \(y=\left(x^{2}+4\right)^{2}\)

$$ \begin{array}{l} \text { . Use the trigonometric identity } \sin 2 x=2 \sin x \cos x\\\ \text { along with the Product Rule to find } D_{x} \sin 2 x \text { . } \end{array} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{1}{2 x}+2 x $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ H(x)=\frac{3}{\sqrt{x-2}} $$

The rate of change of velocity with respect to time is called acceleration. Suppose that the velocity at time \(t\) of a particle is given by \(v(t)=2 t^{2} .\) Find the instantaneous acceleration when \(t=1\) second.

All six sides of a cubical metal box are \(0.25\) inch thick, and the volume of the interior of the box is 40 cubic inches. Use differentials to find the approximate volume of metal used to make the box.

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ \frac{d r}{d x} \text { if } r=\frac{\ln x}{x^{2} \ln x^{2}}+\left(\ln \frac{1}{x}\right)^{3} $$

Find \(d y / d x\). \(y=\sqrt[4]{2 x+1}\)

Suppose that \(g(t)=a t^{2}+b t+c\) and \(g(1)=5\), \(g^{\prime}(1)=3\), and \(g^{\prime \prime}(1)=-4 .\) Find \(a, b\), and \(c .\)

Find the indicated derivative. \(y^{\prime}\) where \(y=(x+\sin x)^{2}\)

Use the trigonometric identity \(\cos 2 x=2 \cos ^{2} x-1\) along with the Product Rule to find \(D_{x} \cos 2 x\).

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{2}{3 x}-\frac{2}{3} $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ H(x)=\sqrt{x^{2}+4} $$

A city is hit by an Asian flu epidemic. Officials estimate that \(t\) days after the beginning of the epidemic the number of persons sick with the flu is given by \(p(t)=120 t^{2}-2 t^{3}\), when \(0 \leq t \leq 40 .\) At what rate is the flu spreading at time \(t=10 ; t=20 ; t=40 ?\)

The outside diameter of a thin spherical shell is 12 feet. If the shell is \(0.3\) inch thick, use differentials to approximate the volume of the region interior to the shell.

Find \(D_{x} y\). $$ y=\tanh (\cot x) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ g^{\prime}(x) \text { if } g(x)=\ln \left(x+\sqrt{x^{2}+1}\right) $$