Question

How do you differentiate `g(z) = z^3sin^2(2z)` using the product rule?

Answer 1

`frac{d}{dz}(z^3sin ^2(2z))=3z^2sin ^2(2z)+4sin (2z)cos (2z)z^3`

Explanation:

`\frac{d}{dz}(z^3sin ^2(2z))`

Applying product rule,
`(fcdot g)^'=f^'cdot g+fcdot g^'`

`f=z^3` ; `g=sin^2(2z)`

`=frac{d}{dz}(z^3)sin ^2(2z)+frac{d}{dz}(sin ^2(2z))z^3`

We know,
`frac{d}{dz}(z^3)=3z^2`
also, `frac{d}{dz}(sin ^2(2z))=4sin (2z)cos (2z)`

So finally,we get,
`3z^2sin ^2(2z)+4sin (2z)cos (2z)z^3`