Calculus Examples

$y = e^{\cos (t)}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = \cos (t)$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\cos (t)$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d t} [\cos (t)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d t} [\cos (t)]$

Step 1.3

Replace all occurrences of $u$ with $\cos (t)$ .

$e^{\cos (t)} \frac{d}{d t} [\cos (t)]$

Step 2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$e^{\cos (t)} (- \sin (t))$

Step 3

Reorder the factors of $e^{\cos (t)} (- \sin (t))$ .

$- e^{\cos (t)} \sin (t)$