Calculus Examples

$f (x) = 3 x^{\ln (x)}$

Step 1

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

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

Step 2

Use the properties of logarithms to simplify the differentiation.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Raise $\ln (x)$ to the power of $1$ .

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

Step 4

Raise $\ln (x)$ to the power of $1$ .

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

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6

Add $1$ and $1$ .

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

Step 7

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 $g (x) = \ln^{2} (x)$ .

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Step 7.1

To apply the Chain Rule, set $u_{1}$ as $\ln^{2} (x)$ .

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

Step 7.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 7.3

Replace all occurrences of $u_{1}$ with $\ln^{2} (x)$ .

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

Step 8

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

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Step 8.1

To apply the Chain Rule, set $u_{2}$ as $\ln (x)$ .

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

Step 8.2

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

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

Step 8.3

Replace all occurrences of $u_{2}$ with $\ln (x)$ .

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

Step 9

Multiply $2$ by $3$ .

$6 e^{\ln^{2} (x)} (\ln (x) \frac{d}{d x} [\ln (x)])$

Step 10

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

$6 e^{\ln^{2} (x)} (\ln (x) \frac{1}{x})$

Step 11

Combine fractions.

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Step 11.1

Combine $\ln (x)$ and $\frac{1}{x}$ .

$6 e^{\ln^{2} (x)} \frac{\ln (x)}{x}$

Step 11.2

Combine $\frac{\ln (x)}{x}$ and $6$ .

$\frac{\ln (x) \cdot 6}{x} e^{\ln^{2} (x)}$

Step 11.3

Combine $\frac{\ln (x) \cdot 6}{x}$ and $e^{\ln^{2} (x)}$ .

$\frac{\ln (x) \cdot 6 e^{\ln^{2} (x)}}{x}$

Step 11.4

Move $6$ to the left of $\ln (x)$ .

$\frac{6 \cdot \ln (x) e^{\ln^{2} (x)}}{x}$

Step 12

Simplify the numerator.

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Step 12.1

Simplify $6 \ln (x)$ by moving $6$ inside the logarithm.

$\frac{\ln (x^{6}) e^{\ln^{2} (x)}}{x}$

Step 12.2

Reorder factors in $\ln (x^{6}) e^{\ln^{2} (x)}$ .

$\frac{e^{\ln^{2} (x)} \ln (x^{6})}{x}$