Question

How do you use the chain rule to differentiate `y=(x^2+3)^4`?

Answer 1

`=>y' = 8x(x^2+3)^3`

Explanation:

Chain rule is:

`(d)/(dx) f(g(x)) = (df(g(x)))/(dg(x)) * (dg(x))/(dx)`

We have:

• `f(g(x)) = (x^2+3)^4`
• `g(x) = (x^2+3)`

So we first compute `(df(g(x)))/(dg(x))`.

`(d(x^2+3)^4)/(d(x^2+3)) = color(blue)(4(x^2+3)^3)`

Now we compute `(dg(x))/(dx)`.

`(d(x^2+3))/(dx) = color(blue)(2x)`

We multiply these two results to get the final solution:

`=>y' = 8x(x^2+3)^3`

Answer 2

`8x(x^2+3)^3`

Explanation:

The chain rule states that,

`dy/dx=dy/(du)*(du)/dx`

Let `u=x^2+3`, then `(du)/dx=2x`.

We have: `y=u^4`, and therefore `dy/(du)=4u^3`.

Multiply these results together to get,

`dy/dx=4u^3*2x`

`=8xu^3`

Now, we undo the substitution `u=x^2+3`, and we get:

`=8x(x^2+3)^3`