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

Nov 1, 2016

$f ' \left(x\right) = \frac{\sin \left(1 + {x}^{2}\right) - 2 {x}^{2} \cos \left(1 + {x}^{2}\right)}{\sin} ^ 2 \left(1 + {x}^{2}\right)$

#### Explanation:

f(x)=x/(sqrt(sin^2(1+x^2))

Use quotient rule and chain rule

$f = x , g = {\left({\left(\sin \left(1 + {x}^{2}\right)\right)}^{2}\right)}^{\frac{1}{2}} = \sin \left(1 + {x}^{2}\right)$

$f ' = 1 , g ' = \cos \left(1 + {x}^{2}\right) \cdot 2 x$

$f ' \left(x\right) = \frac{g f ' - f g '}{g} ^ 2$

$f ' \left(x\right) = \frac{\sin \left(1 + {x}^{2}\right) - 2 {x}^{2} \cos \left(1 + {x}^{2}\right)}{\sin} ^ 2 \left(1 + {x}^{2}\right)$