Calculus Examples

$e^{5 t}$

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 $g (t) = 5 t$ .

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To apply the Chain Rule, set $u$ as $5 t$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d t} [5 t]$

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

$e^{u} \frac{d}{d t} [5 t]$

Replace all occurrences of $u$ with $5 t$ .

$e^{5 t} \frac{d}{d t} [5 t]$

Differentiate.

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Since $5$ is constant with respect to $t$ , the derivative of $5 t$ with respect to $t$ is $5 \frac{d}{d t} [t]$ .

$e^{5 t} (5 \frac{d}{d t} [t])$

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$e^{5 t} (5 \cdot 1)$

Simplify the expression.

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Multiply $5$ by $1$ .

$e^{5 t} \cdot 5$

Move $5$ to the left of $e^{5 t}$ .

$5 e^{5 t}$