Calculus Examples

$f (x) = x^{2 \ln (x)}$

Step 1

Use the properties of logarithms to simplify the differentiation.

Tap for more steps...

Rewrite $x^{2 \ln (x)}$ as $e^{\ln (x^{2 \ln (x)})}$ .

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

Expand $\ln (x^{2 \ln (x)})$ by moving $2 \ln (x)$ outside the logarithm.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Add $1$ and $1$ .

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

Step 6

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

Tap for more steps...

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

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

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

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

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

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

Step 7

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

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

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

Tap for more steps...

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

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

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

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

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

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

Step 9

Multiply $2$ by $2$ .

$e^{2 \ln^{2} (x)} (4 (\ln (x) \frac{d}{d x} [\ln (x)]))$

Step 10

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

$e^{2 \ln^{2} (x)} (4 \ln (x) \frac{1}{x})$

Step 11

Combine fractions.

Tap for more steps...

Combine $\frac{1}{x}$ and $4$ .

$e^{2 \ln^{2} (x)} (\frac{4}{x} \ln (x))$

Combine $\frac{4}{x}$ and $\ln (x)$ .

$e^{2 \ln^{2} (x)} \frac{4 \ln (x)}{x}$

Combine $e^{2 \ln^{2} (x)}$ and $\frac{4 \ln (x)}{x}$ .

$\frac{e^{2 \ln^{2} (x)} (4 \ln (x))}{x}$

Step 12

Simplify the numerator.

Tap for more steps...

Rewrite using the commutative property of multiplication.

$\frac{4 e^{2 \ln^{2} (x)} \ln (x)}{x}$

Multiply $4 e^{2 \ln^{2} (x)} \ln (x)$ .

Tap for more steps...

Reorder $\ln (x)$ and $4$ .

$\frac{4 \cdot \ln (x) e^{2 \ln^{2} (x)}}{x}$

Simplify $4 \cdot \ln (x)$ by moving $4$ inside the logarithm.

$\frac{\ln (x^{4}) e^{2 \ln^{2} (x)}}{x}$

Reorder factors in $\ln (x^{4}) e^{2 \ln^{2} (x)}$ .

$\frac{e^{2 \ln^{2} (x)} \ln (x^{4})}{x}$