Question

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

Answer 1

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):
\[ \frac{dv}{dx}= 15(1+3x)^{4} \]