Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = \ln (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

Combine $\frac{1}{x}$ and $x^{2}$ .

$\frac{\ln (x) (2 x) - \frac{x^{2}}{x}}{\ln^{2} (x)}$

Step 4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{2}$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Raise $x$ to the power of $1$ .

$\frac{\ln (x) (2 x) - \frac{x \cdot x}{x^{1}}}{\ln^{2} (x)}$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{\ln (x) (2 x) - \frac{x \cdot x}{x \cdot 1}}{\ln^{2} (x)}$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) (2 x) - \frac{x \cdot x}{x \cdot 1}}{\ln^{2} (x)}$

Step 4.2.2.4

Rewrite the expression.

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

Step 4.2.2.5

Divide $x$ by $1$ .

$\frac{\ln (x) (2 x) - x}{\ln^{2} (x)}$

Step 5

Simplify the numerator.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \ln (x) x - x}{\ln^{2} (x)}$

Step 5.1.2

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$\frac{\ln (x^{2}) x - x}{\ln^{2} (x)}$

Step 5.2

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

$\frac{x \ln (x^{2}) - x}{\ln^{2} (x)}$