Question

a. use the product rule to find the derivative of the given function
b. find the derivative by expanding the product first

$h(z)=(7 - z^{2})(z^{3}-2z + 3)$

a. use the product rule to find the derivative of the given function. select the correct answer below and fill in the answer box(es) to complete your choice
a. the derivative is $(z^{3}-2z + 3)(\square)$
b. the derivative is $(7 - z^{2})(\square)$
c. the derivative is $(7 - z^{2})(z^{3}-2z + 3)+(\square)$
d. the derivative is $(7 - z^{2})(3z^{2}-2)+(z^{3}-2z + 3)(-2z)$
e. the derivative is $(7 - z^{2})(z^{3}-2z + 3)(\square)$

b. expand the product
$(7 - z^{2})(z^{3}-2z + 3)=-z^{5}+9z^{3}-3z^{2}-14z + 21$ (simplify your answer)

using either approach, $\frac{d}{dz}(7 - z^{2})(z^{3}-2z + 3)=\square$

Explanation:

Step1: Recall product - rule

If $h(z)=f(z)g(z)$, then $h'(z)=f(z)g'(z)+g(z)f'(z)$. Here $f(z)=7 - z^{2}$, $f'(z)=-2z$; $g(z)=z^{3}-2z + 3$, $g'(z)=3z^{2}-2$.

Step2: Apply product - rule

$h'(z)=(7 - z^{2})(3z^{2}-2)+(z^{3}-2z + 3)(-2z)$.

Step3: Expand and simplify

\[

$$\begin{align*} h'(z)&=(7 - z^{2})(3z^{2}-2)+(z^{3}-2z + 3)(-2z)\\ &=21z^{2}-14-3z^{4}+2z^{2}-2z^{4}+4z^{2}-6z\\ &=-5z^{4}+27z^{2}-6z - 14 \end{align*}$$

\]

Answer:

$-5z^{4}+27z^{2}-6z - 14$