What is the Derivative of e^sin(x)?

derivative of e^sin(x)

What is the derivative of e^sin(x)?

What is the derivative of $e^{\sin(x)}$ ?

Solution. Let

h(x) = e^{\sin(x)}

f(u) = e^u

and

g(x) = \sin(x)

. We will use the chain rule:

\begin{align*} h'(x) = f'(g(x))g'(x). \end{align*}

We have that

\frac{d}{dx} e^x = e^x

and

\frac{d}{dx} \sin(x) = \cos(x)

. So we have that

\begin{align*} f'(u) = e^u \quad \text{and} \quad g'(x) = \cos(x). \end{align*}

Therefore, we get

\begin{align*} h'(x) &= f'(g(x))g'(x) \\ &= e^{\sin(x)}\cos(x) \\ &= \cos(x)e^{\sin(x)} \end{align*}

So the derivative of

e^{\sin(x)}

\cos(x)e^{\sin(x)}