# 007A Sample Midterm 1, Problem 4 Detailed Solution

Find the derivatives of the following functions. Do not simplify.

(a)   ${\displaystyle f(x)={\sqrt {x}}(x^{2}+2)}$

(b)   ${\displaystyle g(x)={\frac {x+3}{x^{\frac {3}{2}}+2}}}$ where ${\displaystyle x>0}$

(c)   ${\displaystyle h(x)={\sqrt {x+{\sqrt {x}}}}}$

Background Information:
1. Product Rule
${\displaystyle {\frac {d}{dx}}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)}$
2. Quotient Rule
${\displaystyle {\frac {d}{dx}}{\bigg (}{\frac {f(x)}{g(x)}}{\bigg )}={\frac {g(x)f'(x)-f(x)g'(x)}{(g(x))^{2}}}}$
3. Chain Rule
${\displaystyle {\frac {d}{dx}}(f(g(x)))=f'(g(x))g'(x)}$

Solution:

(a)

Step 1:
Using the Product Rule, we have
${\displaystyle f'(x)=({\sqrt {x}})'(x^{2}+2)+{\sqrt {x}}(x^{2}+2)'.}$
Step 2:
Now, using the Power Rule, we have
${\displaystyle {\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {({\sqrt {x}})'(x^{2}+2)+{\sqrt {x}}(x^{2}+2)'}\\&&\\&=&\displaystyle {{\bigg (}{\frac {1}{2}}x^{-{\frac {1}{2}}}{\bigg )}(x^{2}+2)+{\sqrt {x}}(2x).}\end{array}}}$

(b)

Step 1:
Using the Quotient Rule, we have
${\displaystyle g'(x)={\frac {(x^{\frac {3}{2}}+2)(x+3)'-(x+3)(x^{\frac {3}{2}}+2)'}{(x^{\frac {3}{2}}+2)^{2}}}.}$
Step 2:
Now, using the Power Rule, we have
${\displaystyle {\begin{array}{rcl}\displaystyle {g'(x)}&=&\displaystyle {\frac {(x^{\frac {3}{2}}+2)(x+3)'-(x+3)(x^{\frac {3}{2}}+2)'}{(x^{\frac {3}{2}}+2)^{2}}}\\&&\\&=&\displaystyle {{\frac {(x^{\frac {3}{2}}+2)(1)-(x+3)({\frac {3}{2}}x^{\frac {1}{2}})}{(x^{\frac {3}{2}}+2)^{2}}}.}\end{array}}}$

(c)

Step 1:
First, we write
${\displaystyle h(x)=(x+x^{\frac {1}{2}})^{\frac {1}{2}}.}$
Then, using the Chain Rule, we have
${\displaystyle h'(x)={\frac {1}{2}}(x+x^{\frac {1}{2}})^{-{\frac {1}{2}}}(x+x^{\frac {1}{2}})'.}$
Step 2:
Now, using the Power Rule, we have
${\displaystyle h'(x)={\frac {1}{2}}(x+x^{\frac {1}{2}})^{-{\frac {1}{2}}}{\bigg (}1+{\frac {1}{2}}x^{-{\frac {1}{2}}}{\bigg )}.}$

(a)     ${\displaystyle f'(x)={\bigg (}{\frac {1}{2}}x^{-{\frac {1}{2}}}{\bigg )}(x^{2}+2)+{\sqrt {x}}(2x)}$
(b)     ${\displaystyle g'(x)={\frac {(x^{\frac {3}{2}}+2)(1)-(x+3)({\frac {3}{2}}x^{\frac {1}{2}})}{(x^{\frac {3}{2}}+2)^{2}}}}$
(c)     ${\displaystyle h'(x)={\frac {1}{2}}(x+x^{\frac {1}{2}})^{-{\frac {1}{2}}}{\bigg (}1+{\frac {1}{2}}x^{-{\frac {1}{2}}}{\bigg )}}$