# How do you differentiate f(x)=cscx * sec2x using the product rule?

May 11, 2017

#### Answer:

$\frac{\mathrm{df}}{\mathrm{dx}} = \csc x \left(2 \sec 2 x \tan 2 x - \cot x \sec 2 x\right)$

#### Explanation:

Product rule states if $f \left(x\right) = g \left(x\right) h \left(x\right)$

then $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{dg}}{\mathrm{dx}} \times h \left(x\right) + \frac{\mathrm{dh}}{\mathrm{dx}} \times g \left(x\right)$

Hence as $f \left(x\right) = \csc x \cdot \sec 2 x$

and $\frac{d}{\mathrm{dx}} \csc x = - \csc x \cot x$ and $\frac{d}{\mathrm{dx}} \sec 2 x = \sec 2 x \tan 2 x \times 2$

$\therefore \frac{\mathrm{df}}{\mathrm{dx}} = \left(- \csc x \cot x\right) \times \sec 2 x + \csc x \times \sec 2 x \tan 2 x \times 2$

= $\csc x \left(2 \sec 2 x \tan 2 x - \cot x \sec 2 x\right)$