Calculus Examples

$y = x^{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{x})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Use the properties of logarithms to simplify the differentiation.

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Step 3.1.1

Rewrite $x^{x}$ as $e^{\ln (x^{x})}$ .

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

Step 3.1.2

Expand $\ln (x^{x})$ by moving $x$ outside the logarithm.

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

Step 3.2

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 $g (x) = x \ln (x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x \ln (x)$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x \ln (x)]$

Step 3.2.2

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

$e^{u} \frac{d}{d x} [x \ln (x)]$

Step 3.2.3

Replace all occurrences of $u$ with $x \ln (x)$ .

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

Step 3.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

$e^{x \ln (x)} (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x])$

Step 3.4

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$e^{x \ln (x)} (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.5

Differentiate using the Power Rule.

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Step 3.5.1

Combine $x$ and $\frac{1}{x}$ .

$e^{x \ln (x)} (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.5.2

Cancel the common factor of $x$ .

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Step 3.5.2.1

Cancel the common factor.

$e^{x \ln (x)} (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.5.2.2

Rewrite the expression.

$e^{x \ln (x)} (1 + \ln (x) \frac{d}{d x} [x])$

Step 3.5.3

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

$e^{x \ln (x)} (1 + \ln (x) \cdot 1)$

Step 3.5.4

Multiply $\ln (x)$ by $1$ .

$e^{x \ln (x)} (1 + \ln (x))$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$e^{x \ln (x)} \cdot 1 + e^{x \ln (x)} \ln (x)$

Step 3.6.2

Multiply $e^{x \ln (x)}$ by $1$ .

$e^{x \ln (x)} + e^{x \ln (x)} \ln (x)$

Step 3.6.3

Reorder terms.

$e^{x \ln (x)} \ln (x) + e^{x \ln (x)}$

Step 4

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

$y' = e^{x \ln (x)} \ln (x) + e^{x \ln (x)}$

Step 5

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

$\frac{d y}{d x} = e^{x \ln (x)} \ln (x) + e^{x \ln (x)}$