Question

Solve 10 and label part A and B, make sure you do both a and b

Answer 1

The function given is,

$h(z)=\left(z^3+4z^2+z\right)\left(z-1\right)$

10a) Using the product rule of differentiation

$\begin{gathered} \mathrm{Apply\:the\:Product\:Rule}:\quad \left(f\cdot g\right)'=f\:'\cdot g+f\cdot g' \\ f=z^3+4z^2+z,\:g=z-1 \\ =\frac{d}{dz}\left(z^3+4z^2+z\right)\left(z-1\right)+\frac{d}{dz}\left(z-1\right)\left(z^3+4z^2+z\right) \\ =\left(3z^2+8z+1\right)\left(z-1\right)+1\cdot \left(z^3+4z^2+z\right) \\ \frac{dh}{dz}=4z^3+9z^2-6z-1 \end{gathered}$

10b) Using the expanding product first before differentiating

$\begin{gathered} h(z)=(z^{3}+4z^{2}+z)(z-1) \\ h(z)=z^4+3z^3-3z^2-z \\ \frac{dh}{dz}=4(z)^{4-1}+3\times3(z^{3-1})-2\times3(z^{2-1})-1=4z^3+9z^2-6z-1 \\ \therefore\frac{dh}{dz}=4z^3+9z^2-6z-1 \end{gathered}$

Hence, from the results, it is verified that the answer agrees with part a.