# Differentiation by the chain rule

## Homework Statement

Find the derivative of the following:

Y= x^3(5x-1)^4

## The Attempt at a Solution

4(3x^2(5x-1)^3)(4(3x^2(3(5x-1)^2)(2(5x-1)(5)

## Answers and Replies

Dick
Science Advisor
Homework Helper
That doesn't look like the chain rule to me. Apply the product rule first.

i tried bringing down the 4Th exponent and then subtract it by one to get three, then leaving the inside alone ( 5x-1) at the same time taking the derivative of 3x^2. after that i got confused and got a wrong answer.

jambaugh
Science Advisor
Gold Member
$y = f(x)g(x)$ where $f(x)= x^3$ and $g(x)=(5x-1)^4$
So you'll first need to apply the product rule... as you do you'll need the derivative of g.

$g(x) = P\circ L (x) = P( L(x))$ where $P(x) = x^4$ and $L(x)=5x - 1$. As a composition you need to apply the chain rule. (P for power, L for linear).

If you'd rather use the Leibniz notation form of the chain rule: $\frac{du}{dx} = \frac{du}{dv} \frac{dv}{dx}$ then let u=g(x) = P(v) with v = L(x).

Dick
Science Advisor
Homework Helper
Your function is f*g where f=x^3 and g=(5x-1)^4, right? The product rule says the derivative of f*g is f'*g+f*g', also right? Now you just need to find f' and g'. Finding the derivative of g' is where you need the chain rule.