Question

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

Answer 1

`y = -4x +(9pi)/4`

Explanation:

We have using implicit differentiation:

`1(sin(16x))(dy/dx) + y(16cos(16x)) = 1(cos(2y)) - 2xsin(2y)(dy/dx)`

`sin(16x)(dy/dx) + 2xsin(2y)(dy/dx) = cos(2y) - 16ycos(16x)`

`dy/dx = (cos(2y) - 16ycos(16x))/(sin(16x) + 2xsin(2y))`

We now find the slope of the tangent at that point.

`dy/dx|_((pi/2, pi/4)) = (cos(2(pi/4)) - 16(pi/4)cos(16(pi/2)))/(sin(16(pi/2)) + 2(pi/2)sin(2(pi/4))`

`dy/dx|_((pi/2, pi/4)) = (0 - 4pi)/(0 + pi)`

`dy/dx|_((pi/2, pi/4)) = -4`

Now for the equation

`y - pi/4 = -4(x- pi/2)`

`y = -4x + 2pi + pi/4 = -4x + (9pi)/4`

Hopefully this helps!