Derivative of cos(e^t)

The calculator will find the derivative of

\cos{\left(e^{t} \right)}

, with steps shown.

Your Input

Find $\frac{d}{dt} \left(\cos{\left(e^{t} \right)}\right)$ .

Solution

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

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

{\color{red}\left(\frac{d}{dt} \left(\cos{\left(e^{t} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dt} \left(e^{t}\right)\right)}

The derivative of the cosine is $\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$ :

{\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dt} \left(e^{t}\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dt} \left(e^{t}\right)

Return to the old variable:

- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dt} \left(e^{t}\right) = - \sin{\left({\color{red}\left(e^{t}\right)} \right)} \frac{d}{dt} \left(e^{t}\right)

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

- \sin{\left(e^{t} \right)} {\color{red}\left(\frac{d}{dt} \left(e^{t}\right)\right)} = - \sin{\left(e^{t} \right)} {\color{red}\left(e^{t}\right)}

Thus, $\frac{d}{dt} \left(\cos{\left(e^{t} \right)}\right) = - e^{t} \sin{\left(e^{t} \right)}$ .

Answer

$\frac{d}{dt} \left(\cos{\left(e^{t} \right)}\right) = - e^{t} \sin{\left(e^{t} \right)}$ A

Derivative of cos⁡(et)\cos{\left(e^{t} \right)}cos(et)

Your Input

Solution

Answer

Derivative of $\cos{\left(e^{t} \right)}$