What is the slope of the tangent at x=pi for f(x)=e^sinx

Question

What is the slope of the tangent at x=pi

for f(x)=e^sinx

Answer 1

For f(x) or y equal to e^{sin x}, obtain dy/dx = d(sin x)/dx times e^{sin x} or (cos x)(e^{sin x}).

Since the slope of the line tangent to the graph of e^{sin x} (for any value of x for which e^{sin x} is defined)

equals (cos x) × (e^{sin x}), it follows that the slope of the tangent to y = e^{sin x} at x = pi (or π or 3.1415926535897932384626433832795) is given by cos π × e^{sin π} or -1 × e⁰ or -1×1 or -1.

Answer 2

We first need to find the formula for f ' for the given function. We will need to use the Chain Rule since we have a composition of functions in e^{sin x}. So f '(x) = e^sinx · cos x. Now if we evaluate this at x = π, we have e^{sin π} · cos π. Since sin π = 0, e⁰= 1, and since cos π = -1, our final product is 1(-1) or -1 for the slope of f(x) at x = π.