Derivative of e^(sin(x))

The calculator will find the derivative of

e^{\sin{\left(x \right)}}

, with steps shown.

Your Input

Find $\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right)$ .

Solution

The function $e^{\sin{\left(x \right)}}$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = e^{u}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$ :

{\color{red}\left(\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}

The derivative of the exponential is $\frac{d}{du} \left(e^{u}\right) = e^{u}$ :

{\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)

Return to the old variable:

e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = e^{{\color{red}\left(\sin{\left(x \right)}\right)}} \frac{d}{dx} \left(\sin{\left(x \right)}\right)

The derivative of the sine is $\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$ :

e^{\sin{\left(x \right)}} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = e^{\sin{\left(x \right)}} {\color{red}\left(\cos{\left(x \right)}\right)}

Thus, $\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right) = e^{\sin{\left(x \right)}} \cos{\left(x \right)}$ .

Answer

$\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right) = e^{\sin{\left(x \right)}} \cos{\left(x \right)}$ A

Derivative of esin⁡(x)e^{\sin{\left(x \right)}}esin(x)

Your Input

Solution

Answer

Derivative of $e^{\sin{\left(x \right)}}$