Calculus Examples

$f (t) = 10 e^{\frac{1}{2} t}$

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

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

$\frac{d}{d t} [10 e^{\frac{t}{2}}]$

Step 1.2

Since $10$ is constant with respect to $t$ , the derivative of $10 e^{\frac{t}{2}}$ with respect to $t$ is $10 \frac{d}{d t} [e^{\frac{t}{2}}]$ .

$10 \frac{d}{d t} [e^{\frac{t}{2}}]$

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

Tap for more steps...

Step 2.1

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

$10 (\frac{d}{d u} [e^{u}] \frac{d}{d t} [\frac{t}{2}])$

Step 2.2

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

$10 (e^{u} \frac{d}{d t} [\frac{t}{2}])$

Step 2.3

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

$10 (e^{\frac{t}{2}} \frac{d}{d t} [\frac{t}{2}])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

$10 e^{\frac{t}{2}} (\frac{1}{2} \frac{d}{d t} [t])$

Step 3.2

Simplify terms.

Tap for more steps...

Step 3.2.1

Combine $\frac{1}{2}$ and $10$ .

$\frac{10}{2} e^{\frac{t}{2}} \frac{d}{d t} [t]$

Step 3.2.2

Combine $\frac{10}{2}$ and $e^{\frac{t}{2}}$ .

$\frac{10 e^{\frac{t}{2}}}{2} \frac{d}{d t} [t]$

Step 3.2.3

Cancel the common factor of $10$ and $2$ .

Tap for more steps...

Step 3.2.3.1

Factor $2$ out of $10 e^{\frac{t}{2}}$ .

$\frac{2 (5 e^{\frac{t}{2}})}{2} \frac{d}{d t} [t]$

Step 3.2.3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (5 e^{\frac{t}{2}})}{2 (1)} \frac{d}{d t} [t]$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{2 (5 e^{\frac{t}{2}})}{2 \cdot 1} \frac{d}{d t} [t]$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{5 e^{\frac{t}{2}}}{1} \frac{d}{d t} [t]$

Step 3.2.3.2.4

Divide $5 e^{\frac{t}{2}}$ by $1$ .

$5 e^{\frac{t}{2}} \frac{d}{d t} [t]$

Step 3.3

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

$5 e^{\frac{t}{2}} \cdot 1$

Step 3.4

Multiply $5$ by $1$ .

$5 e^{\frac{t}{2}}$