Calculus Examples

$\frac{x^{4}}{x - 12}$

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 $f (x) = x^{4}$ and $g (x) = x - 12$ .

$\frac{(x - 12) \frac{d}{d x} [x^{4}] - x^{4} \frac{d}{d x} [x - 12]}{{(x - 12)}^{2}}$

Step 2

Differentiate.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{(x - 12) (4 x^{3}) - x^{4} \frac{d}{d x} [x - 12]}{{(x - 12)}^{2}}$

Step 2.2

Move $4$ to the left of $x - 12$ .

$\frac{4 \cdot (x - 12) x^{3} - x^{4} \frac{d}{d x} [x - 12]}{{(x - 12)}^{2}}$

Step 2.3

By the Sum Rule, the derivative of $x - 12$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 12]$ .

$\frac{4 (x - 12) x^{3} - x^{4} (\frac{d}{d x} [x] + \frac{d}{d x} [- 12])}{{(x - 12)}^{2}}$

Step 2.4

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

$\frac{4 (x - 12) x^{3} - x^{4} (1 + \frac{d}{d x} [- 12])}{{(x - 12)}^{2}}$

Step 2.5

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$\frac{4 (x - 12) x^{3} - x^{4} (1 + 0)}{{(x - 12)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $1$ and $0$ .

$\frac{4 (x - 12) x^{3} - x^{4} \cdot 1}{{(x - 12)}^{2}}$

Step 2.6.2

Multiply $- 1$ by $1$ .

$\frac{4 (x - 12) x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(4 x + 4 \cdot - 12) x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{4 x \cdot x^{3} + 4 \cdot - 12 x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.3.1.1.1

Move $x^{3}$ .

$\frac{4 (x^{3} x) + 4 \cdot - 12 x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3.3.1.1.2

Multiply $x^{3}$ by $x$ .

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Step 3.3.1.1.2.1

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

$\frac{4 (x^{3} x^{1}) + 4 \cdot - 12 x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3.3.1.1.2.2

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

$\frac{4 x^{3 + 1} + 4 \cdot - 12 x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3.3.1.1.3

Add $3$ and $1$ .

$\frac{4 x^{4} + 4 \cdot - 12 x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3.3.1.2

Multiply $4$ by $- 12$ .

$\frac{4 x^{4} - 48 x^{3} - x^{4}}{{(x - 12)}^{2}}$

Step 3.3.2

Subtract $x^{4}$ from $4 x^{4}$ .

$\frac{3 x^{4} - 48 x^{3}}{{(x - 12)}^{2}}$

Step 3.4

Factor $3 x^{3}$ out of $3 x^{4} - 48 x^{3}$ .

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Step 3.4.1

Factor $3 x^{3}$ out of $3 x^{4}$ .

$\frac{3 x^{3} (x) - 48 x^{3}}{{(x - 12)}^{2}}$

Step 3.4.2

Factor $3 x^{3}$ out of $- 48 x^{3}$ .

$\frac{3 x^{3} (x) + 3 x^{3} (- 16)}{{(x - 12)}^{2}}$

Step 3.4.3

Factor $3 x^{3}$ out of $3 x^{3} (x) + 3 x^{3} (- 16)$ .

$\frac{3 x^{3} (x - 16)}{{(x - 12)}^{2}}$