Question

What is the derivative of `(1-x^2)/(x^3)`?

Answer 1

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

Explanation:

Split up the fraction and rewrite it with negative exponents:

`(1-x^2)/x^3=1/x^3-x^2/x^3=x^-3-x^-1`

Now, differentiate `x^-3-x^-1` using the power rule, which states that the derivative of `x^n=nx^(n-1)`. Thus:

`d/dx(x^-3-x^-1)=(-3)x^(-3-1)-(-1)x^(-1-1)`

`=-3x^-4+x^-2`

This is a valid final answer, but we can also write it without negative exponents by multiplying it by `x^4/x^4`:

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