Find an equation for the tangent line to the implicit curve at a point.

Question

sin(x−y) =xcos(y+π/4) at the point (π/4, π/4)

Answer 1

Implicit differentiation gives:

[cos(x-y)](1-y') = -x[sin(y+π/4)]y' + cos(y+π/4)

The above result was achieved as follows:

On the left use the derivative of sine followed by the chain rule.

On the right of the equal sign, use the product rule, which also requires the derivative of cosine along with the chain rule.

Since you want to determine the value of y' when x=π/4 and y=π/4, it is easier to just substitute those values now (rather than attempting to isolate y'). So...

cos(0)(1-y') = -π/4 sin (π/2)y' + cos(π/2)

1-y'=-π/4y'

1=y'-π/4y'

1=y'(1-π/4)

y' = 1/(1-π/4)

Of course that value is the slope of the tangent line at the given point. From there use point-slope to write the equation of the tangent line.

https://www.desmos.com/calculator/glo6wtwrct

The above graph shows the curve and the tangent line.