Equation of the tangent

Question

Determine the equation of the tangent to the curve: cos (x y) = ye^x -pi/2

at the point (x; y) = (0,pi/2) you will need to use implicit differentiation.

Answer 1

(0, π/2) is not a solution of cos(xy) = ye^x - π/2, but is a solution of the equation cos(x+y) = ye^x - π/2.

cos(x+y) = ye^x - π/2

-sin(x + y)(1 + dy/dx) = (dy/dx)e^x + e^xy

At the point (0, π/2), we have -sin(π/2)(1 + dy/dx) = (dy/dx)(1) + π/2

-1 - dy/dx = dy/dx + π/2

2(dy/dx) = -1 - π/2

dy/dx = -1/2 - π/4

Tangent line: y - π/2 = (-1/2 - π/4)(x - 0)