Calculus Examples

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

Step 1

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 $g (x) = \frac{2}{3} x$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

Combine $\frac{2}{3}$ and $x$ .

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

Step 2.2

Combine $\frac{2}{3}$ and $x$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $\frac{2 e^{\frac{2 x}{3}}}{3}$ by $1$ .

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