Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin(16x) = x cos(2y), (π/2, π/4)

Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

y sin(16x) = x cos(2y),    (π /2, π /4)

Answer 1

This graph shows 3 things:

1. The graph of this implicitly defined relation.
2. The point (pi/2, pi/4) does indeed lie on the graph
3. The point-slope form for the equation of the tangent line.

The graph: 

https://www.desmos.com/calculator/svlwczzkqr

How did I get the slope = -4?

I implicitly differentiated with respect to x and plugged x=pi/2, y=pi/4 and solved for y'.

Since you want the slope at a specific point you can substitute before you solve for y' (if you want).

On both sides of the equation you use the product rule followed by the chain rule.

ycos(16x)(16)+ sin(16x)y' = x (-sin(2y))(2y') + cos(2y)

If you want a formula for y' in terms of x and y: terms with y' on the left, terms without y' on the right (of the equal sign). Factor out the y' and divide both sides by the coefficient of y'. Easier to just substitute x=pi/2, y=pi/4 and evaluate right now. THEN solve for y' based on those substitutions.