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

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Answer 1 · 2016-05-14T19:32:34

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

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 $x^4/x^4$ :

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

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