Question

If `f(x)=g(x^6)/x^2` , and if g is a differentiable function, find an expression for `f''(x)` ?

Answer 1

`f''(x)=(-4x^5(g'(x^6))+x^2(g''(x^6))-2x(g(x^6))-2(g(x^6)))/x^4`

Explanation:

.

`f(x)=(g(x^6))/x^2`

Using the Quotient Rule:

`f'(x)=(x^2(g'(x^6))-(g(x^6))(2x))/x^4`

`f''(x)=(x^4((2x)(g'(x^6))+x^2(g''(x^6)-(g'(x^6))(2x)-(g(x^6))(2))-4x^3(x^2(g'(x^6))-(g(x^6))(2x))))/x^8`

`f''(x)=(2x^5(g'(x^6))+x^6(g''(x^6))-2x^5(g'(x^6))-2x^4(g(x^6))-4x^9(g'(x^6))-2x^5(g(x^6)))/x^8`

`f''(x)=(cancelcolor(red)(2x^5(g'(x^6)))+x^6(g''(x^6))cancelcolor(red)(-2x^5(g'(x^6)))-2x^4(g(x^6))-4x^9(g'(x^6))-2x^5(g(x^6)))/x^8`

`f''(x)=(x^4(-4x^5(g'(x^6))+x^2(g''(x^6))-2x(g(x^6))-2(g(x^6))))/(x^4(x^4)`

`f''(x)=(cancelcolor(red)(x^4)(-4x^5(g'(x^6))+x^2(g''(x^6))-2x(g(x^6))-2(g(x^6))))/(cancelcolor(red)(x^4)(x^4)`

`f''(x)=(-4x^5(g'(x^6))+x^2(g''(x^6))-2x(g(x^6))-2(g(x^6)))/x^4`