# If f(x)= cot5 x  and g(x) = 2x^2 -1 , how do you differentiate f(g(x))  using the chain rule?

Mar 18, 2016

$\frac{\mathrm{df}}{\mathrm{dx}} - 4 x {\csc}^{2} \left(10 {x}^{2} - 5\right)$

#### Explanation:

If $f \left(x\right) = \cot 5 x$ and $g \left(x\right) = 2 {x}^{2} - 1$

$f \left(g \left(x\right)\right) = \cot \left(5 \cdot \left(2 {x}^{2} - 1\right)\right)$

As according to chain rule, $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dg}} \times \frac{\mathrm{dg}}{\mathrm{dx}}$

$\frac{\mathrm{df}}{\mathrm{dx}} = - {\csc}^{2} \left(5 \cdot \left(2 {x}^{2} - 1\right)\right) \times \frac{d}{\mathrm{dx}} \left(2 {x}^{2} - 1\right)$

= -csc^2(10x^2-5)xx4x)

= $- 4 x {\csc}^{2} \left(10 {x}^{2} - 5\right)$