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

To implicitly differentiate this curve, we simply take each term in the equation and differentiate it with respect to x

Differentiating cos(xy):

(cos(xy))' = -sin(xy) * (xy)'

= -sin(xy) * (x' * y + y' * x)

= -sin(xy) * (y + y' * x)

Differentiating y*e^x:

(y*e^x)' = y' * e^x + y * e^x

And the derivative of -pi/2 = 0

We want to isolate all terms containing y' on one side of the equation, such that we can write y' as a function of x and y. Then we can just plug in x and y to calculate y'.

y' * (-sin(xy) * x) - y * sin(xy) = e^x * y + e^x * y'

-> y'(e^x + x * sin(xy)) = -y * (e^x + sin(xy))

-> y' = -y (e^x + x * sin(xy)) / (e^x + sin(xy))

Now we plug in x = 0, y = pi/2

-> y' = -pi/2 * (1 + 0) / (1 + 0)

-> y' = -pi/2