Calculus Examples

$y = e^{3 \sqrt{x}}$

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

$y = e^{3 x^{\frac{1}{2}}}$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{3 x^{\frac{1}{2}}})$

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 3 x^{\frac{1}{2}}$ .

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To apply the Chain Rule, set $u$ as $3 x^{\frac{1}{2}}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x^{\frac{1}{2}}]$

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

$e^{u} \frac{d}{d x} [3 x^{\frac{1}{2}}]$

Replace all occurrences of $u$ with $3 x^{\frac{1}{2}}$ .

$e^{3 x^{\frac{1}{2}}} \frac{d}{d x} [3 x^{\frac{1}{2}}]$

Differentiate.

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Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{1}{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$e^{3 x^{\frac{1}{2}}} (3 \frac{d}{d x} [x^{\frac{1}{2}}])$

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

$e^{3 x^{\frac{1}{2}}} (3 (\frac{1}{2} x^{\frac{1}{2} - 1}))$

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

$e^{3 x^{\frac{1}{2}}} (3 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

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

$e^{3 x^{\frac{1}{2}}} (3 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Combine the numerators over the common denominator.

$e^{3 x^{\frac{1}{2}}} (3 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$e^{3 x^{\frac{1}{2}}} (3 (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Subtract $2$ from $1$ .

$e^{3 x^{\frac{1}{2}}} (3 (\frac{1}{2} x^{\frac{- 1}{2}}))$

Move the negative in front of the fraction.

$e^{3 x^{\frac{1}{2}}} (3 (\frac{1}{2} x^{- \frac{1}{2}}))$

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

$e^{3 x^{\frac{1}{2}}} (3 \frac{x^{- \frac{1}{2}}}{2})$

Combine $3$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$e^{3 x^{\frac{1}{2}}} \frac{3 x^{- \frac{1}{2}}}{2}$

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

$e^{3 x^{\frac{1}{2}}} \frac{3}{2 x^{\frac{1}{2}}}$

Combine $e^{3 x^{\frac{1}{2}}}$ and $\frac{3}{2 x^{\frac{1}{2}}}$ .

$\frac{e^{3 x^{\frac{1}{2}}} \cdot 3}{2 x^{\frac{1}{2}}}$

Move $3$ to the left of $e^{3 x^{\frac{1}{2}}}$ .

$\frac{3 e^{3 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}}}$

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3 e^{3 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}}}$

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3 e^{3 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}}}$