Question

How do you differentiate `f(x)=x^7sqrt(4x^2+7)` using the product rule?

Answer 1

`f'(x) = 7x^6(4x^2 + 7)^(1/2) + (4x^8)/(4x^2 + 7)^(1/2)`

Explanation:

Let `f(x) = g(x) xx h(x)`

The derivative of `f(x)`, by the product rule, is given by `f'(x) = g'(x) xx h(x) + g(x) xx h'(x)`

We must therefore differentiate both `g(x) and h(x)`.

`g'(x) = 7 xx x^(7 - 1)`

`g'(x) = 7x^6`

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We differentiate `h(x)` using the chain rule.

Letting `y = u^(1/2)` and `u =4x^2 + 7`

`dy/dx = 1/2u^(-1/2) xx 8x`

`dy/dx = (8x)/(2(4x^2 + 7)^(1/2))`

`dy/dx = (4x)/(4x^2 + 7)^(1/2)`

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We now have all the information we need to apply the product rule.

`f'(x) = 7x^6 xx (4x^2 + 7)^(1/2) + (4x)/(4x^2 + 7)^(1/2) xx x^7`

`f'(x) = 7x^6(4x^2 + 7)^(1/2) + (4x^8)/(4x^2 + 7)^(1/2)`

This can be simplified further, but I'll leave the algebra to you.

Hopefully this helps!