Question

How do you differentiate `f(x)= (2x^2-5)(4x^2+5)` using the product rule?

Answer 1

`f'(x) = (4x)(8x^2-5)`

Explanation:

If you are studying maths, then you should learn the Product Rule for Differentiation, and practice how to use it:

`d/dx(uv)=u(dv)/dx+(du)/dxv`, or, `(uv)' = (du)v + u(dv)`

I was taught to remember the rule in words; "The first times the derivative of the second plus the derivative of the first times the second ".

So with `f(x) = (2x^2-5)(4x^2+5)` we have;

`{ ("Let "u = 2x^2-5, => , (du)/dx =4x), ("And "v = 4x^2+5, =>, (dv)/dx = 8x ) :}`

`\ \ \ \ \ d/dx(uv) = u(dv)/dx + (du)/dxv`
`:. d/dx(uv) = (2x^2-5)(8x) + (4x)(4x^2+5)`
`:. d/dx(uv) = (4x){2(2x^2-5) + (4x^2+5)}`
`:. d/dx(uv) = (4x)(4x^2-10+4x^2+5)`
`:. d/dx(uv) = (4x)(8x^2-5)`