Calculus Examples

$y = 2 e^{\cos (t)}$

Step 1

Since $2$ is constant with respect to $t$ , the derivative of $2 e^{\cos (t)}$ with respect to $t$ is $2 \frac{d}{d t} [e^{\cos (t)}]$ .

$2 \frac{d}{d t} [e^{\cos (t)}]$

Step 2

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 $g (t) = \cos (t)$ .

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Step 2.1

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

$2 (\frac{d}{d u} [e^{u}] \frac{d}{d t} [\cos (t)])$

Step 2.2

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

$2 (e^{u} \frac{d}{d t} [\cos (t)])$

Step 2.3

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

$2 (e^{\cos (t)} \frac{d}{d t} [\cos (t)])$

Step 3

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

$2 e^{\cos (t)} (- \sin (t))$

Step 4

Multiply $- 1$ by $2$ .

$- 2 e^{\cos (t)} \sin (t)$