# How do you use the chain rule to differentiate f(x)=sin(1/(x^2+1))?

Mar 26, 2018

See below

#### Explanation:

$f \left(x\right) = S \in \left(\frac{1}{{x}^{2} + 1}\right)$

$f \left(x\right) = S \in \left(u\right)$

$f ' \left(x\right) = C o s \left(u\right) \times u '$

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

(Substitute again to include the inner function)

$v = {x}^{2} + 1$
$v ' = 2 x$

$u ' = - 1 \left(v '\right) {\left(v\right)}^{-} 2 = - 1 \left(2 x\right) {\left({x}^{2} + 1\right)}^{-} 2 = \frac{- 2 x}{{x}^{2} + 1} ^ 2$

Hence:

$f ' \left(x\right) = \left(\frac{- 2 x}{{x}^{2} + 1} ^ 2\right) C o s \left(\frac{1}{{x}^{2} + 1}\right)$