Question

How do you find the derivative of `((x^3)-(7/x))^-2`?

Answer 1

`(-2x(3x^4+7))/(x^4-7)^3`

Explanation:

Rewrite & then use a combination of the power rule & chain rule :
`y = (x^3-7x^-1)^-2`

`y' = -2(x^3-7x^-1)^-3(3x^2+7x^-2)`

Simplify by making all exponents positive & find common denominators:

`y' = (-2(3x^2+7/x^2))/(x^3-7/x)^3 = (-2((3x^4+7)/x^2))/((x^4-7)/x)^3`

Simplify by cubing both numerator & denominator of the denominator term and multiplying reciprocals:
`y' = -2((3x^4+7)/x^2)(x^3)/(x^4-7)^3`

Simplify and rearrange to get the final answer:
`(-2x(3x^4+7))/(x^4-7)^3`