Question

How do you differentiate `y = (6e^(-7x)+2x)^2` using the chain rule?

Answer 1

`y'=-504e^(-14x)+12e^(-7x)-84xe^(-7x)+4x`

Explanation:

To differentiate the given function `y` using chain rule let:
`f(x)=x^2` and
`g(x)=6e^(-7x)+2x`
So, `y=f(g(x))`

To differentiate `y=f(g(x))` we have to use chain rule as follows:
Then `y'=(f(g(x)))'=f'(g(x))*g'(x)`

Let's find `f'(x)` and `g'(x)`
`f'(x)=2x`
`g'(x)=-7*6e^(-7x)+2=-42e^(-7x)+2`

`y'=(f(g(x)))'=f'(g(x))*g'(x)`

`y'=2(6e^(-7x)+2x)*(-42e^(-7x)+2)`
`y'=2(-252e^(-14x)+12e^(-7x)-84xe^(-7x)+4x)`
`y'=-504e^(-14x)+12e^(-7x)-84xe^(-7x)+4x`