Calculus Examples

${(5 x)}^{x}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $5$ out of $5 x$ .

$\frac{d}{d x} [{(5 (x))}^{x}]$

Step 1.2

Apply the product rule to $5 (x)$ .

$\frac{d}{d x} [5^{x} x^{x}]$

Step 2

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 $f (x) = 5^{x}$ and $g (x) = x^{x}$ .

$5^{x} \frac{d}{d x} [x^{x}] + x^{x} \frac{d}{d x} [5^{x}]$

Step 3

Use the properties of logarithms to simplify the differentiation.

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Step 3.1

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

$5^{x} \frac{d}{d x} [e^{\ln (x^{x})}] + x^{x} \frac{d}{d x} [5^{x}]$

Step 3.2

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

$5^{x} \frac{d}{d x} [e^{x \ln (x)}] + x^{x} \frac{d}{d x} [5^{x}]$

Step 4

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 4.1

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

$5^{x} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x \ln (x)]) + x^{x} \frac{d}{d x} [5^{x}]$

Step 4.2

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

$5^{x} (e^{u} \frac{d}{d x} [x \ln (x)]) + x^{x} \frac{d}{d x} [5^{x}]$

Step 4.3

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

$5^{x} (e^{x \ln (x)} \frac{d}{d x} [x \ln (x)]) + x^{x} \frac{d}{d x} [5^{x}]$

Step 5

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

$5^{x} (e^{x \ln (x)} (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x])) + x^{x} \frac{d}{d x} [5^{x}]$

Step 6

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

$5^{x} (e^{x \ln (x)} (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])) + x^{x} \frac{d}{d x} [5^{x}]$

Step 7

Differentiate using the Power Rule.

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Step 7.1

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

$5^{x} (e^{x \ln (x)} (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])) + x^{x} \frac{d}{d x} [5^{x}]$

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

$5^{x} (e^{x \ln (x)} (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])) + x^{x} \frac{d}{d x} [5^{x}]$

Step 7.2.2

Rewrite the expression.

$5^{x} (e^{x \ln (x)} (1 + \ln (x) \frac{d}{d x} [x])) + x^{x} \frac{d}{d x} [5^{x}]$

Step 7.3

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

$5^{x} (e^{x \ln (x)} (1 + \ln (x) \cdot 1)) + x^{x} \frac{d}{d x} [5^{x}]$

Step 7.4

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

$5^{x} (e^{x \ln (x)} (1 + \ln (x))) + x^{x} \frac{d}{d x} [5^{x}]$

Step 8

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$5^{x} (e^{x \ln (x)} (1 + \ln (x))) + x^{x} (5^{x} \ln (5))$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$5^{x} (e^{x \ln (x)} \cdot 1 + e^{x \ln (x)} \ln (x)) + x^{x} (5^{x} \ln (5))$

Step 9.2

Apply the distributive property.

$5^{x} (e^{x \ln (x)} \cdot 1) + 5^{x} (e^{x \ln (x)} \ln (x)) + x^{x} (5^{x} \ln (5))$

Step 9.3

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

$5^{x} e^{x \ln (x)} + 5^{x} (e^{x \ln (x)} \ln (x)) + x^{x} (5^{x} \ln (5))$

Step 9.4

Reorder terms.

$5^{x} e^{x \ln (x)} + 5^{x} e^{x \ln (x)} \ln (x) + 5^{x} x^{x} \ln (5)$