compute the derivative of the given function in two different ways. (h(x)=\frac{x^{3}+3x^{2}}{2x^{2}}) a…

compute the derivative of the given function in two different ways. (h(x)=\frac{x^{3}+3x^{2}}{2x^{2}}) a. use the quotient rule, (\frac{f}{g}=\frac{gcdot f - fcdot g}{g^{2}}). (fill in each blank, then simplify.) (h(x)=\frac{(square)cdot(square)-(square)cdot(square)}{square}=square) b. use algebra first to simplify (h), then differentiate without the quotient rule. (h(x)=square) note: you can earn partial credit on this problem. preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining.

compute the derivative of the given function in two different ways. (h(x)=\frac{x^{3}+3x^{2}}{2x^{2}}) a. use the quotient rule, (\frac{f}{g}=\frac{gcdot f - fcdot g}{g^{2}}). (fill in each blank, then simplify.) (h(x)=\frac{(square)cdot(square)-(square)cdot(square)}{square}=square) b. use algebra first to simplify (h), then differentiate without the quotient rule. (h(x)=square) note: you can earn partial credit on this problem. preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining.

Answer

Explanation:

Step1: Identify f and g for Quotient Rule

Let $f(x)=x^{3}+3x^{2}$ and $g(x) = 2x^{2}$. Then $f'(x)=3x^{2}+6x$ and $g'(x)=4x$.

Step2: Apply Quotient Rule

$h'(x)=\frac{(2x^{2})\cdot(3x^{2}+6x)-(x^{3}+3x^{2})\cdot(4x)}{(2x^{2})^{2}}$ $=\frac{6x^{4}+12x^{3}-4x^{4}-12x^{3}}{4x^{4}}$ $=\frac{2x^{4}}{4x^{4}}=\frac{1}{2}$

Step3: Simplify h(x) first for part b

$h(x)=\frac{x^{3}+3x^{2}}{2x^{2}}=\frac{1}{2}x+\frac{3}{2}$

Step4: Differentiate simplified h(x)

$h'(x)=\frac{1}{2}$

Answer:

a. $h'(x)=\frac{(2x^{2})\cdot(3x^{2}+6x)-(x^{3}+3x^{2})\cdot(4x)}{(2x^{2})^{2}}=\frac{1}{2}$ b. $h'(x)=\frac{1}{2}$