Question

How do you use the Product Rule to find the derivative of `(x^2)(x^3+4)`?

Answer 1

The explanation is given below.

Explanation:

Product rule for differentiation where product of functions is given.

If `f(x)` and `g(x)` are two functions then their product would be `f(x)*g(x)`

The derivative `(f(x)*g(x))'` is given by the product rule

`(f(x)g(x))' = f(x)g'(x)+g(x)f'(x)`

Derivative rules

`d/dx (x^n) = nx^(n-1)`
`d/dx(f(x)+g(x)) = d/dx(f(x)) + d/dx(g(x))`
`d/dx(c) = 0` Derivative of constant is zero.

Now coming to our problem

`y=(x^2)(x^3+4)`

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

`f'(x) = d/dx(x^2) => f'(x) = 2x`
`g'(x) = d/dx(x^3+4) = d/dx (x^3) + d/dx(4)`
`g'(x) = 3x^2 + 0`
`g'(x) = 3x^2`

Using product rule

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

`= x^2(3x^2) + (x^3+4)(2x)`

`= 3x^4 + 2x^4+8x` simplifying

`=5x^4+8x` Answer