How do you differentiate Y= x squared(1+3x)power 5? Do you use the product rule or chain rule?

Question

phi · Answer

Use both the product rule and the chain rule

let  $u= x^{2}$ and $v= (1+3x)^{5}$
$$ \frac{d\ u\cdot v}{dx}= v\frac{du}{dx} +u \frac{dv}{dx} $$

You can use the chain rule to find  $ \frac{dv}{dx}$
let $w= (1+3x)$ and  $v= w^{5}$
$$ \frac{dv}{dx}= \frac{dv}{dw}\cdot \frac{dw}{dx} $$
This gives us
$$ \frac{dv}{dx}=  5w^{4} \cdot 3= 15w^{4} $$
replacing w with (1+3x):
$$ \frac{dv}{dx}= 15(1+3x)^{4} $$