Chapter 4

Differentiate with respect to the independent variable. $$ f(x)=\sqrt{x}\left(x^{4}-5 x^{2}\right) $$

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

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\frac{1}{x-3} $$

Assume that \(x\) and \(y\) are differentiable functions of \(t\). Find \(\frac{d y}{d t}\) when \(x^{2}+y^{2}=1, \frac{d x}{d t}=2\) for \(x=\frac{1}{2}\), and \(y>0 .\)

Use the identity $$ \cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta $$ and the definition of the derivative to show that $$ \frac{d}{d x} \cos x=-\sin x $$

Bacterial Growth Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N(0) 2^{t} $$ where \(N(0)\) is the population size at time \(0 .\) Find the rate of growth \(d N / d t .\) Express your solution in terms of \(N(t) .\) Show that the growth rate of the population is proportional to the population size.

Differentiate with respect to the independent variable. $$ f(x)=\sqrt{3 x}\left(x^{2}-1\right) $$

Let \(f(x)=\ln x .\) We know that \(f^{\prime}(x)=\frac{1}{x} .\) We will use this fact and the definition of derivatives to show that $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e$$ (a) Use the definition of the derivative to show that $$f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{\ln (1+h)}{h}$$ (b) Show that (a) implies that $$\ln \left[\lim _{h \rightarrow 0}(1+h)^{1 / h}\right]=1$$ (c) Set \(h=\frac{1}{n}\) in (b) and let \(n \rightarrow \infty\). Show that this implies that $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e$$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\frac{3-x}{3+x} $$

Assume that \(x\) and \(y\) are differentiable functions of \(t\). Find \(\frac{d y}{d t}\) when \(y^{2}=x^{2}-x^{4}, \frac{d x}{d t}=1\) for \(x=\frac{1}{2}\), and \(y>0\).

Use the quotient rule to show that $$ \frac{d}{d x} \cot x=-\csc ^{2} x $$

Bacterial Growth Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N(0) 2^{t} $$ where \(N(0)\) is the population size at time \(0 .\) Find the per capita growth rate.

Differentiate with respect to the independent variable. $$ f(x)=\frac{\sqrt{5 x\left(1+x^{2}\right)}}{\sqrt{2}} $$

Assume that \(f(x)\) is differentiable with respect to \(x\). Show that $$\frac{d}{d x} \ln \left[\frac{f(x)}{x}\right]=\frac{f^{\prime}(x)}{f(x)}-\frac{1}{x}$$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\frac{x-1}{x+1} $$

Assume that \(x\) and \(y\) are differentiable functions of \(t\). Find \(\frac{d y}{d t}\) when \(x^{2} y=1\) and \(\frac{d x}{d t}=3\) for \(x=2\).

Use the quotient rule to show that $$ \frac{d}{d x} \sec x=\sec x \tan x $$

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

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=2 x^{x} $$

In Problems \(63-70\), find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=x^{2} $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\left|x^{2}-3\right| $$

Assume that \(u\) and \(v\) are differentiable functions of \(t\). Find \(\frac{d u}{d t}\) when \(u^{2}+v^{3}=12, \frac{d v}{d t}=2\) for \(v=2\), and \(u>0\).

Use the quotient rule to show that $$ \frac{d}{d x} \csc x=-\csc x \cot x $$

Fish Recruitment Model The following model is used in the fisheries literature to describe the recruitment of fish as a function of the size of the parent stock: If we denote the number of recruits by \(R\) and the size of the parent stock by \(P\), then $$ R(P)=\alpha P e^{-\beta P}, \quad P \geq 0 $$ where \(\alpha\) and \(\beta\) are positive constants. (a) Sketch the graph of the function \(R(P)\) when \(\beta=1\) and \(\alpha=2\). (b) Differentiate \(R(P)\) with respect to \(P\). (c) Find all the points on the curve that have a horizontal tangent.

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

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=(2 x)^{2 x} $$

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=2-x^{2} $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\left|2 x^{2}-1\right| $$

Assume that the side length \(x\) and the volume \(V=x^{3}\) of a cube are differentiable functions of \(t .\) Express \(d V / d t\) in terms of \(d x / d t\)

Find the derivative of $$ f(x)=\sin \sqrt{x^{2}+1} $$

Von Bertalanffy Growth Model The growth of fish can be described by the von Bertalanffy growth function $$ L(x)=L_{\infty}-\left(L_{\infty}-L_{0}\right) e^{-k x} $$ where \(x\) denotes the age of the fish and \(k, L_{\infty}\), and \(L_{0}\) are positive constants. (a) Set \(L_{0}=1\) and \(L_{\infty}=10 .\) Graph \(L(x)\) for \(k=1.0\) and \(k=0.1\). (b) Interpret \(L_{\infty}\) and \(L_{0}\). (c) Compare the graphs for \(k=0.1\) and \(k=1.0\). According to which graph do fish reach \(L=5\) more quickly? (d) Show that $$ \frac{d}{d x} L(x)=k\left(L_{\infty}-L(x)\right) $$ That is, \(d L / d x \propto L_{\infty}-L\). What does this proportionality say about how the rate of growth changes with age? (e) The constant \(k\) is the proportionality constant in (d). What does the value of \(k\) tell you about how quickly a fish grows?

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

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=(\ln x)^{x} $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ f(x)=\left\\{\begin{array}{cl} x & \text { for } x \leq 0 \\ x+1 & \text { for } x>0 \end{array}\right. $$

Assume that the radius \(r\) and the area \(A=\pi r^{2}\) of a circle are differentiable functions of \(t .\) Express \(d A / d t\) in terms of \(d r / d t\).

Find the derivative of $$ f(x)=\cos \sqrt{x^{2}+1} $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the radioactive decay rate of the material is \(0.2 /\) day. Find the differential equation for the radioactive decay function \(W(t)\).

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

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=(\ln x)^{3 x} $$

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=4 x+2 x^{2} $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ f(x)=\left\\{\begin{array}{cl} 2 x & \text { for } x \leq 1 \\ x+2 & \text { for } x>1 \end{array}\right. $$

Assume that the radius \(r\) and the surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t .\) Express \(d S / d t\) in terms of \(d r / d t\)

Find the derivative of $$ f(x)=\sin \sqrt{3 x^{3}+3 x} $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the radioactive decay rate of the material is \(4 /\) day. Find the differential equation for the radioactive decay function \(W(t)\).

Differentiate with respect to the independent variable. $$ g(s)=\frac{s^{1 / 3}-1}{s^{2 / 3}-1} $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{\ln x} $$

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=3 x^{3}-x^{2} $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ \text { 8. } f(x)=\left\\{\begin{array}{cl} x^{2} & \text { for } x \leq-1 \\ 2-x^{2} & \text { for } x>-1 \end{array}\right. $$

Assume that the radius \(r\) and the volume \(V=\frac{4}{3} \pi r^{3}\) of a sphere are differentiable functions of \(t .\) Express \(d V / d t\) in terms of \(d r / d t\)

Find the derivative of $$ f(x)=\cos \sqrt{1-4 x^{4}} $$