# How do you use chain rule to find the derivative of y = 2 csc^3(sqrt(x)) ?

Mar 1, 2015

You can easily illustrate the chain rule using Leibniz notation

Let $y = 2 {u}^{3}$ then $\frac{\mathrm{dy}}{\mathrm{du}} = 6 {u}^{2}$

Let $u = \csc \left(w\right)$ then $\frac{\mathrm{du}}{\mathrm{dw}} = - \csc \left(w\right) \cot \left(w\right)$

Let $w = \sqrt{x}$ then $\frac{\mathrm{dw}}{\mathrm{dx}} = \frac{1}{2 \sqrt{x}}$

Now the chain rule is

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dw}} \frac{\mathrm{dw}}{\mathrm{dx}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 6 {u}^{2} \left(- \csc \left(w\right) \cot \left(w\right)\right) \frac{1}{2 \sqrt{x}}$

Remember that $u = \csc \left(w\right)$ and $w = \sqrt{x}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 6 {\csc}^{2} \left(\sqrt{x}\right) \left(- \csc \left(\sqrt{x}\right) \cot \left(\sqrt{x}\right)\right) \left(\frac{1}{2 \sqrt{x}}\right)$

Some simplifying

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 3 {\csc}^{3} \left(\sqrt{x}\right) \cot \left(\sqrt{x}\right)}{\sqrt{x}}$