Calculus Examples

$4 e^{\frac{t}{2}}$

Since $4$ is constant with respect to $t$ , the derivative of $4 e^{\frac{t}{2}}$ with respect to $t$ is $4 \frac{d}{d t} [e^{\frac{t}{2}}]$ .

$4 \frac{d}{d t} [e^{\frac{t}{2}}]$

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 $g (t) = \frac{t}{2}$ .

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To apply the Chain Rule, set $u$ as $\frac{t}{2}$ .

$4 (\frac{d}{d u} [e^{u}] \frac{d}{d t} [\frac{t}{2}])$

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

$4 (e^{u} \frac{d}{d t} [\frac{t}{2}])$

Replace all occurrences of $u$ with $\frac{t}{2}$ .

$4 (e^{\frac{t}{2}} \frac{d}{d t} [\frac{t}{2}])$

Differentiate.

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Since $\frac{1}{2}$ is constant with respect to $t$ , the derivative of $\frac{t}{2}$ with respect to $t$ is $\frac{1}{2} \frac{d}{d t} [t]$ .

$4 e^{\frac{t}{2}} (\frac{1}{2} \frac{d}{d t} [t])$

Simplify terms.

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Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} e^{\frac{t}{2}} \frac{d}{d t} [t]$

Combine $\frac{4}{2}$ and $e^{\frac{t}{2}}$ .

$\frac{4 e^{\frac{t}{2}}}{2} \frac{d}{d t} [t]$

Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4 e^{\frac{t}{2}}$ .

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

Cancel the common factors.

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Factor $2$ out of $2$ .

$\frac{2 (2 e^{\frac{t}{2}})}{2 (1)} \frac{d}{d t} [t]$

Cancel the common factor.

$\frac{2 (2 e^{\frac{t}{2}})}{2 \cdot 1} \frac{d}{d t} [t]$

Rewrite the expression.

$\frac{2 e^{\frac{t}{2}}}{1} \frac{d}{d t} [t]$

Divide $2 e^{\frac{t}{2}}$ by $1$ .

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

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

$2 e^{\frac{t}{2}} \cdot 1$

Multiply $2$ by $1$ .

$2 e^{\frac{t}{2}}$