Question

How do you differentiate `f(x)= (1-3x^2)^4* (1-x+7x^2)^4` using the product rule?

Answer 1

`f'(x)=4(1-x+7x^2)^3(1-3x^2)^3(-84x^3+9x^2+8x-1)`

Explanation:

`f(x)=(1-3x^2)^4*(1-x+7x^2)^4`
Using product rule,
`f'(x)=(1-3x^2)^4*d/dx(1-x+7x^2)^4+(1-x+7x^2)^4*d/dx(1-3x^2)^4`
Now,
`d/dx(1-x+7x^2)^4=4(1-x+7x^2)^3*d/dx(1-x+7x^2)=4(1-x+7x^2)^3(14x-1)`
and,
`d/dx(1-3x^2)^4=4(1-3x^2)^3*d/dx(1-3x^2)=4(1-3x^2)^3(-6x)`
Put these two values in the expression for f'(x) and extract the common terms to get,
`f'(x)=4(1-x+7x^2)^3(1-3x^2)^3[(1-3x^2)(14x-1)-6x(1-x+7x^2)]`
Further solving the equation we get, `f'(x)=4(1-x+7x^2)^3(1-3x^2)^3(-84x^3+9x^2+8x-1)`
which is the final answer