Chapter 4

Differentiate the functions with respect to the independent variable. \(g(r)=2^{-3 \sin r}\)

Differentiate with respect to the independent variable. \(f(x)=\frac{1-4 x^{3}}{1-x}\)

Find the normal line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=1-3 x^{2}, \text { at } x=-2 $$

Find the derivative with respect to the independent variable. $$ h(s)=\sin ^{3} s+\cos ^{3} s $$

Differentiate the functions with respect to the independent variable. \(g(r)=3^{r^{1 / 5}}\)

Differentiate with respect to the independent variable. \(f(x)=\frac{3 x^{2}-2 x+1}{2 x+1}\)

Find the normal line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=\sqrt{3} x^{4}-2 \sqrt{3} x^{2}, \text { at } x=-\sqrt{3} $$

Find the derivative with respect to the independent variable. $$ f(x)=\left(2 x^{3}-x\right) \cos ^{2} x $$

Differentiate the functions with respect to the independent variable. \(g(r)=4^{r^{1 / 4}}\)

Differentiate with respect to the independent variable. \(f(x)=\frac{x^{4}+2 x-1}{5 x^{2}-2 x+1}\)

Find the normal line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=-2 x^{2}-x, \text { at } x=0 $$

Find the derivative with respect to the independent variable. $$ f(x)=\frac{\sin (2 x)}{1+x^{2}} $$

Compute the limits. \(\lim _{h \rightarrow 0} \frac{e^{2 h}-1}{h}\)

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \left|x^{2}-3\right| $$

Differentiate with respect to the independent variable. \(f(x)=\frac{3-x^{3}}{1-x}\)

Find the normal line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=x^{3}-3, \text { at } x=1 $$

Find the derivative with respect to the independent variable. $$ f(x)=\frac{1+\cos (3 x)}{2 x^{3}-x} $$

Compute the limits. \(\lim _{h \rightarrow 0} \frac{e^{5 h}-1}{3 h}\)

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\log \left(2 x^{2}-1\right) $$

Differentiate with respect to the independent variable. \(f(x)=\frac{1+2 x^{2}-4 x^{4}}{3 x^{3}-5 x^{5}}\)

Find the normal line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=1-\pi x^{2}, \text { at } x=-1 $$

Find the derivative with respect to the independent variable. $$ f(x)=\tan \frac{1}{x} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\log \left(1-x^{2}\right) $$

Differentiate with respect to the independent variable. \(h(t)=\frac{t^{2}-3 t+1}{t+1}\)

Find the tangent line to $$ f(x)=a x^{2} $$ at \(x=1\). Assume that \(a\) is a positive constant.

Find the derivative with respect to the independent variable. $$ f(x)=\sec \left(\frac{1}{1+x^{2}}\right) $$

Compute the limits. \(\lim _{h \rightarrow 0} \frac{2^{h}-1}{h}\)

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\log \left(3 x^{3}-x+2\right) $$

Differentiate with respect to the independent variable. \(h(t)=\frac{3-t^{2}}{(t-1)^{2}}\)

Find the tangent line to $$ f(x)=a x^{3}-2 a x $$ at \(x=-1\). Assume that \(a\) is a positive constant.

Find the derivative with respect to the independent variable. $$ f(x)=\frac{\cos x^{2}}{\cos ^{2} x} $$

Find the equation for the tangent to the curve \(y=2^{x}\) at the point \((1,2)\).

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\log \left(x^{3}-3 x\right) $$

Differentiate with respect to the independent variable. \(f(s)=\frac{4-2 s^{2}}{1-s}\)

Find the tangent line to $$ f(x)=\frac{a x^{2}}{a^{2}+2} $$ at \(x=2\). Assume that \(a\) is a positive constant.

Find the derivative with respect to the independent variable. $$ f(x)=\frac{\csc \left(3-x^{2}\right)}{1-x^{2}} $$

Find the equation for the tangent to the curve \(y=\exp \left[x^{2}\right]\) at the point \(\left(2, e^{4}\right)\).

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\log \left(\sqrt[3]{\tan x^{2}}\right) $$

Differentiate with respect to the independent variable. \(f(s)=\frac{2 s^{3}-4 s^{2}+3 s-4}{\left(s^{2}-3\right)^{2}}\)

Find the tangent line to $$ f(x)=\frac{x^{2}+x}{a+1} $$ at \(x=a\). Assume that \(a\) is a positive constant.

Find the points on the curve \(y=\sin \left(\frac{\pi}{3} x\right)\) that have a horizontal tangent.

Population Growth Suppose that the population size at time \(t\) is $$ N(t)=e^{2 t}, \quad t \geq 0 $$ (a) What is the population size at time 0 ? (b) Show that $$ \frac{d N}{d t}=2 N $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(u)=\log _{3}\left(3+u^{4}\right) $$

Differentiate with respect to the independent variable. \(f(x)=\sqrt{x}(x-1)\)

Find the normal line to $$ f(x)=a x^{3} $$ at \(x=-1\). Assume that \(a\) is a positive constant.

Find the points on the curve \(y=\cos ^{2} x\) that have a horizontal tangent.

Suppose that the population size at time \(t\) is $$ N(t)=N_{0} e^{r t}, \quad t \geq 0 $$ where \(N_{0}\) is a positive constant and \(r\) is a real number. (a) What is the population size at time \(0 ?\) (b) Show that $$ \frac{d N}{d t}=r N $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ g(s)=\log _{5}\left(3^{s}-2\right) $$

Differentiate with respect to the independent variable. \(f(x)=\sqrt{x}\left(x^{4}-x^{2}\right)\)

Find the normal line to $$ f(x)=a x^{2}-3 a x $$ at \(x=2 .\) Assume that \(a\) is a positive constant.