Chapter 3, Problem 68RE

a.

To determine

To find: The first derivative of the following combinations at the given value of x.

  $3f(x)-g(x),x=-1.$

Answer to Problem 68RE

First derivative of the $3f(x)-g(x),x=-1.$ is 5.

Explanation of Solution

Given information: Suppose that function f and g and their first derivative have the

following values at $x=-1\text{and}x=0.$

  

Calculation:

  $\begin{array}{l}(3f(x)-g(x))\text{'}=3\times f\text{'}(x)-g\text{'}(x).\\ \text{Substituting}x=-1.\\ 3\times f\text{'}(-1)-g\text{'}(-1)=3\times 2-1=6-1=5\end{array}$

b.

To determine

To find: The first derivative of the following combinations at the given value of x.

  $f(x)g(x),x=0.$

Answer to Problem 68RE

First derivative of the $f(x)g(x),x=0.$ is 2.

Explanation of Solution

Given information: Suppose that function f and g and their first derivative have the

following values at $x=-1\text{and}x=0.$

  

Calculation:

  $\begin{array}{l}(f(x).g(x))\text{'}=f\text{'}(x)g(x)+f(x)g\text{'}(x).\\ \text{Substituting}x=0.\\ f\text{'}(0)g(0)+f(0)g\text{'}(0)=(-2)\times (-3)+(4)\times (-1)=6-4=2\end{array}$

c.

To determine

To find: The first derivative of the following combinations at the given value of x.

  $f(x)g(x),x=-1.$

Answer to Problem 68RE

First derivative of the $f(x)g(x),x=-1.$ is - 2.

Explanation of Solution

Given information: Suppose that function f and g and their first derivative have the

following values at $x=-1\text{and}x=0.$

  

Calculation:

  $\begin{array}{l}(f(x).g(x))\text{'}=f\text{'}(x)g(x)+f(x)g\text{'}(x).\\ \text{Substituting}x=-1.\\ f\text{'}(-1)g(-1)+f(-1)g\text{'}(-1)=(0)\times (1)+(-1)\times (2)=0-2=-2\end{array}$

d.

To determine

To find: The first derivative of the following combinations at the given value of x.

  $f(x)/g(x),x=0.$

Answer to Problem 68RE

First derivative of the $f(x)/g(x),x=0.$ is 10/9.

Explanation of Solution

Given information: Suppose that function f and g and their first derivative have the

following values at $x=-1\text{and}x=0.$

  

Calculation:

  $\begin{array}{l}$ (f(x)/g(x))\text{'}=(f\text{'}(x)g(x)-f(x)g\text{'}(x))/{(g(x))}^2.$\text{Substituting}x=0.\\ $ (f\text{'}(0)g(0)-f(0)g\text{'}(0))/{(g(0))}^2=((-2)\times (-3)-(-1)\times (4))/{(-3)}^2=(6+4)/9=10/9$\end{array}$

f.

To determine

To find: The first derivative of the following combinations at the given value of x.

  $f(x)/(g(x)+2),x=0.$

Answer to Problem 68RE

First derivative of the $f(x)/(g(x)+2),x=0.$ is 6.

Explanation of Solution

Given information: Suppose that function f and g and their first derivative have the

following values at $x=-1\text{and}x=0.$

  

Calculation:

  $\begin{array}{l}$ (f(x)/(g(x)+2))\text{'}=(f\text{'}(x)(g(x)+2)-f(x)g\text{'}(x))/{(g(x)+2)}^2.$\text{Substituting}x=0.\\ $ (f\text{'}(0)(g(0)+2)+f(0)g\text{'}(0))/{(g(0)+2)}^2=((-2)\times (-3+2)-(-1)\times (4))/{(-3+2)}^2=(2+4)/1=6$\end{array}$