Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin 8x = x cos 2y, (p/2, p/4)

Question

please help 

Answer 1

Implicit differentiation takes the derivative of every variable in an equation with respect to an independent variable.  So, if we start with the equation and differentiate it, we get, using the Product Rule on both sides:

 

dydx sin(8x) + 8y cos(8x) = cos (2y) - 2x sin (2y) dydx

 

Collecting terms gives:

 

dydx [sin (8x) + 2x sin (2y)] = cos (2y) - 8y cos (8x)

 

Now, check the contents of the parentheses on both sides, to make sure you're not dividing anything by zero here.

With x = pi/2 and y = pi/4,

 

dydx [ sin (8[pi/2]) + 2 [pi/2] sin (2[pi/4])] = cos (2[pi/4]) - 8 [pi/4] cos (8[pi/2])

 

dydx [ sin (4pi) + pi sin (pi/2)] = cos (pi/2) - 2pi cos (4pi)

 

or

 

dydx [ 0 + pi] = 0 - 2pi

 

so

 

dydx = -2pi/pi = -2

 

and we can use the Point Slope representation of a line to write the equation of the tangent line as:

 

[ y - pi/4] = -2 [x - pi/2]

 

or

 

y = 5pi/4 - 2x