# How do you use implicit differentiation to find an equation of the tangent line to the curve at the given point ysin16x=xcos2y and (pi/2, pi/4)?

May 4, 2017

$y + 4 x = \frac{9 \pi}{4}$

#### Explanation:

The slope of the tangent to any curve is given by $\frac{\mathrm{dy}}{\mathrm{dx}}$

As $y \sin 16 x = x \cos 2 y$ and differentiating

$\frac{\mathrm{dy}}{\mathrm{dx}} \sin 16 x + 16 y \cos 16 x = 1 \times \cos 2 y - 2 x \sin 2 y \frac{\mathrm{dy}}{\mathrm{dx}}$

or $\frac{\mathrm{dy}}{\mathrm{dx}} \sin 16 x + 2 x \sin 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = \cos 2 y - 16 y \cos 16 x$

or $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\cos 2 y - 16 y \cos 16 x}{\sin 16 x + 2 x \sin 2 y}$

Hence at $\left(\frac{\pi}{2} , \frac{\pi}{4}\right)$ slope is

$\frac{\cos 2 \left(\frac{\pi}{4}\right) - 16 \left(\frac{\pi}{4}\right) \cos \left(4 \pi\right)}{\sin \left(8 \pi\right) + 2 \frac{\pi}{2} \sin \left(\frac{\pi}{2}\right)}$

= $\frac{0 - 4 \pi}{0 + \pi} = - 4$

As tangent is at $\left(\frac{\pi}{2} , \frac{\pi}{4}\right)$, its equation is

$\left(y - \frac{\pi}{4}\right) = - 4 \left(x - \frac{\pi}{2}\right)$ or $y + 4 x = \frac{9 \pi}{4}$