What is the derivative of this function f(x) = sin (1/x^2)?

Jul 13, 2016

$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = \frac{- 2 \cos \left(\frac{1}{x} ^ 2\right)}{x} ^ 3$

Explanation:

This is a simple chain rule problem. It is a little easier if we write the equation as:
$f \left(x\right) = \sin \left({x}^{-} 2\right)$

This reminds us that $\frac{1}{x} ^ 2$ can be differentiated the same way as any polynomial, by dropping the exponent and and reducing it by one.

The application of the chain rule looks like:
$\frac{d}{\mathrm{dx}} \sin \left({x}^{-} 2\right) = \cos \left({x}^{-} 2\right) \left(\frac{d}{\mathrm{dx}} {x}^{-} 2\right)$
$= \cos \left({x}^{-} 2\right) \left(- 2 {x}^{-} 3\right)$
$= \frac{- 2 \cos \left(\frac{1}{x} ^ 2\right)}{x} ^ 3$