# What is the derivative of 1/(1 + x^4)^(1/2)?

Apr 30, 2015

$\frac{- 2 {x}^{3}}{1 + {x}^{4}} ^ \left(\frac{3}{2}\right)$

Solution:

rewrite: $\frac{1}{1 + {x}^{4}} ^ \left(\frac{1}{2}\right) = {\left(1 + {x}^{4}\right)}^{- \frac{1}{2}}$

Now use the power rule and the chain rule (a combination sometimes called the generalized power rule)

The derivative is:

$- \frac{1}{2} {\left(1 + {x}^{4}\right)}^{- \frac{3}{2}} \left(4 {x}^{3}\right) = \frac{- 2 {x}^{3}}{1 + {x}^{4}} ^ \left(\frac{3}{2}\right)$

Apr 30, 2015

$y = {\left(1 + {x}^{4}\right)}^{- \frac{1}{2}} \Rightarrow$
$y ' = - \frac{1}{2} {\left(1 + {x}^{4}\right)}^{- \frac{1}{2} - 1} \cdot 4 {x}^{3} = - \frac{2 {x}^{3}}{\sqrt{{\left(1 + {x}^{4}\right)}^{3}}}$.