Chapter 4, Problem 66RE

a.

To determine

Find the first derivativeof the following combination at the given value of x .

Answer to Problem 66RE

  $(\sqrt{x}f(x))\text{'}=-\frac{13}{2}$ .

Explanation of Solution

Given information: Suppose that a function fand its first derivative have the following

values at x =1.

  

  $\sqrt{x}f(x),x=1.$

Calculation:

  $\begin{array}{l}(\sqrt{x}f(x))\text{'}=\frac{f(x)}{2\sqrt{x}}+\sqrt{x}f\text{'}(x)\\ x=1,f(x)=-3,f\text{'}(x)=\frac{1}{5}\\ (\sqrt{x}f(x))\text{'}=\frac{-3}{2\sqrt{1}}+\sqrt{1}.\frac{1}{5}\\ (\sqrt{x}f(x))\text{'}=-\frac{13}{2}.\end{array}$

b.

To determine

Find the first derivativeof the following combination at the given value of x .

Answer to Problem 66RE

  $(\sqrt{f(x)})\text{'}=-\frac{1}{3}.$

Explanation of Solution

Given information: Suppose that a function fand its first derivative

have the following values at x =1.

  

  $\sqrt{f(x)},x=0$

Calculation:

  $\begin{array}{l}(\sqrt{f(x)})\text{'}=\frac{f\text{'}(x)}{2\sqrt{f(x)}}\\ x=0.f(x)=9,f\text{'}(x)=-2\\ (\sqrt{f(x)})\text{'}=\frac{-2}{2\sqrt{9}}=-\frac{1}{3}\\ (\sqrt{f(x)})\text{'}=-\frac{1}{3}.\end{array}$

c.

To determine

Find the first derivativeof the following combination at the given value of x .

Answer to Problem 66RE

  $f\text{'}(\sqrt{1}).\frac{1}{2\sqrt{1}}=\frac{1}{10}$ .

Explanation of Solution

Given information: Suppose that a function fand its first derivative

have the following values at x =1.

  

  $f(\sqrt{x}),x=1$

Calculation:

  $\begin{array}{l}f(\sqrt{x})=f(x^{\frac{1}{2}})\\ \frac{dy}{dx}[f(\sqrt{x})]=f\text{'}(x^{\frac{1}{2}}).\frac{dy}{dx}(x^{\frac{1}{2}})\\ \frac{dy}{dx}[f(\sqrt{x})]=f\text{'}(x^{\frac{1}{2}}).\frac{1}{2}{x}^{\frac{-1}{2}}\\ \frac{dy}{dx}[f(\sqrt{x})]=f\text{'}(\sqrt{x}).\frac{1}{2\sqrt{x}}\\ f\text{'}(\sqrt{1}).\frac{1}{2\sqrt{1}}=f\text{'}(1).\frac{1}{2}=\frac{1}{5}.\frac{1}{2}=\frac{1}{10}\\ f\text{'}(\sqrt{1}).\frac{1}{2\sqrt{1}}=\frac{1}{10}\end{array}$

d.

To determine

Find the first derivativeof the following combination at the given value of x .

Answer to Problem 66RE

  $f\text{'}(1-5\tan 0).(-5{\sec }^{2}0)=-1$

Explanation of Solution

Given information: Suppose that a function fand its first derivative

have the following values at x =1

  

  $f(1-5\tan x),x=0.$

Calculation:

  $\begin{array}{l}\frac{dy}{dx}[f(1-5\tan x)]=f\text{'}(1-5\tan x).\frac{dy}{dx}(1-5\tan x)\\ \frac{dy}{dx}[f(1-5\tan x)]=f\text{'}(1-5\tan x).(-5{\sec }^{2}x).\\ x=0.\\ f\text{'}(1-5\tan 0).(-5{\sec }^{2}0)=f\text{'}(1-5(0)).(-5(1))\\ f\text{'}(1-5\tan 0).(-5{\sec }^{2}0)=f\text{'}(1).(-5)=\frac{1}{5}(-5)=-1\\ f\text{'}(1-5\tan 0).(-5{\sec }^{2}0)=-1.\end{array}$

e.

To determine

Find the first derivativeof the following combination at the given value of x .

Answer to Problem 66RE

  $\frac{dy}{dx}(\frac{f(0)}{2+\cos 0})=\frac{-2}{3}$

Explanation of Solution

Given information: Suppose that a function fand its first derivative

have the following values at x =1.

  

  $\frac{f(x)}{2+\cos x},x=0.$

Calculation:

  $\begin{array}{l}\frac{dy}{dx}(\frac{f(x)}{2+\cos x})=\frac{f\text{'}(x).2+\cos x-(f(x).(-\sin x))}{{(2+\cos x)}^2}\\ \frac{dy}{dx}(\frac{f(x)}{2+\cos x})=\frac{2+\cos x(-2)-9(-\sin x)}{4+4\cos x+{\cos }^{2}x}\\ \frac{dy}{dx}(\frac{f(0)}{2+\cos 0})=\frac{2+\cos 0(-2)-9(-\sin 0)}{4+4\cos 0+{\cos }^{2}0}\\ \frac{dy}{dx}(\frac{f(0)}{2+\cos 0})=\frac{-2}{3}.\end{array}$

f.

To determine

Find the first derivativeof the following combination at the given value of x .

Answer to Problem 66RE

  $f\text{'}(1)=-12.$

Explanation of Solution

Given information: Suppose that a function fand its first derivative

have the following values at x =1.

  

  $10\sin (\frac{\pi x}{2})f^2(x),x=1.$

Calculation:

  $\begin{array}{l}f(x)=10\sin (\frac{\pi x}{2})f^2(x)\\ f\text{'}(x)=10\sin (\frac{\pi x}{2})(2f(x)).f\text{'}(x)+f^2(x)10\cos \frac{\pi x}{2}\\ f\text{'}(1)=20\sin (\frac{\pi }{2})f(1).f\text{'}(1)+f^2(1)10\cos \frac{\pi }{2}\\ f\text{'}(1)=20(1)(-3)(\frac{1}{5})+f^2(1)10.(0)\\ f\text{'}(1)=-12.\end{array}$