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 \\ =(d)/(dz)\left(z^3+4z^2+z\right)\left(z-1\right)+(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) \\ (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 \\ (dh)/(dz)=4(z)^(4-1)+3*3(z^(3-1))-2*3(z^(2-1))-1=4z^3+9z^2-6z-1 \\ \therefore(dh)/(dz)=4z^3+9z^2-6z-1 \end{gathered}`

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