Calculus Examples

$2^{\sqrt{t}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = 2^{t}$ and $g (t) = t^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $t^{\frac{1}{2}}$ .

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $2$ .

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

Step 2.3

Replace all occurrences of $u$ with $t^{\frac{1}{2}}$ .

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{2}$ .

$2^{t^{\frac{1}{2}}} \ln (2) (\frac{1}{2} t^{\frac{1}{2} - 1})$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2^{t^{\frac{1}{2}}} \ln (2) (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 5

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

$2^{t^{\frac{1}{2}}} \ln (2) (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 6

Combine the numerators over the common denominator.

$2^{t^{\frac{1}{2}}} \ln (2) (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}})$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

$2^{t^{\frac{1}{2}}} \ln (2) (\frac{1}{2} t^{\frac{1 - 2}{2}})$

Step 7.2

Subtract $2$ from $1$ .

$2^{t^{\frac{1}{2}}} \ln (2) (\frac{1}{2} t^{\frac{- 1}{2}})$

Step 8

Move the negative in front of the fraction.

$2^{t^{\frac{1}{2}}} \ln (2) (\frac{1}{2} t^{- \frac{1}{2}})$

Step 9

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

$2^{t^{\frac{1}{2}}} \ln (2) \frac{t^{- \frac{1}{2}}}{2}$

Step 10

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

$\frac{t^{- \frac{1}{2}} \cdot 2^{t^{\frac{1}{2}}}}{2} \ln (2)$

Step 11

Combine $\frac{t^{- \frac{1}{2}} \cdot 2^{t^{\frac{1}{2}}}}{2}$ and $\ln (2)$ .

$\frac{t^{- \frac{1}{2}} \cdot 2^{t^{\frac{1}{2}}} \ln (2)}{2}$

Step 12

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2^{t^{\frac{1}{2}}} \ln (2)}{2 t^{\frac{1}{2}}}$

Step 13

Factor $2$ out of $2^{t^{\frac{1}{2}}} \ln (2)$ .

$\frac{2 (2^{t^{\frac{1}{2}} - 1} \ln (2))}{2 t^{\frac{1}{2}}}$

Step 14

Cancel the common factors.

Tap for more steps...

Step 14.1

Factor $2$ out of $2 t^{\frac{1}{2}}$ .

$\frac{2 (2^{t^{\frac{1}{2}} - 1} \ln (2))}{2 (t^{\frac{1}{2}})}$

Step 14.2

Cancel the common factor.

$\frac{2 (2^{t^{\frac{1}{2}} - 1} \ln (2))}{2 t^{\frac{1}{2}}}$

Step 14.3

Rewrite the expression.

$\frac{2^{t^{\frac{1}{2}} - 1} \ln (2)}{t^{\frac{1}{2}}}$