Let f(x)= e^x cos x find the gradient of the normal to the curve of f at x = π

Question

Help me Calculas Problem

I can't understan gradient of normal

Answer 1

Let f(x) = e^x cos(x)

then the gradient of f(x) is the derivative of y wrt x or

f'(x) = e^x(cos(x) - sin(x))

When x = pi, the gradient becomes

f'(pi) = e^pi(cos(pi)  - sin(pi)) = -e^pi

Since the product of the gradients of two perpendicular lines is equal to -1, the gradient of the normal to the curve is equal to -1/f'(x)

Therefore the gradient of the normal to the curve at the point x = pi is

-1 / -e^pi = e^-pi