Calculus Examples

$y = 4 x^{- 3} + 6 x^{- 2}$

Step 1

By the Sum Rule, the derivative of $4 x^{- 3} + 6 x^{- 2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{- 3}] + \frac{d}{d x} [6 x^{- 2}]$ .

$\frac{d}{d x} [4 x^{- 3}] + \frac{d}{d x} [6 x^{- 2}]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{- 3}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{- 3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{- 3}]$ .

$4 \frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [6 x^{- 2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$4 (- 3 x^{- 4}) + \frac{d}{d x} [6 x^{- 2}]$

Step 2.3

Multiply $- 3$ by $4$ .

$- 12 x^{- 4} + \frac{d}{d x} [6 x^{- 2}]$

Step 3

Evaluate $\frac{d}{d x} [6 x^{- 2}]$ .

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Step 3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{- 2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{- 2}]$ .

$- 12 x^{- 4} + 6 \frac{d}{d x} [x^{- 2}]$

Step 3.2

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

$- 12 x^{- 4} + 6 (- 2 x^{- 3})$

Step 3.3

Multiply $- 2$ by $6$ .

$- 12 x^{- 4} - 12 x^{- 3}$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$- 12 x^{- 3} - 12 x^{- 4}$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 12 \frac{1}{x^{3}} - 12 x^{- 4}$

Step 4.2.2

Combine $- 12$ and $\frac{1}{x^{3}}$ .

$\frac{- 12}{x^{3}} - 12 x^{- 4}$

Step 4.2.3

Move the negative in front of the fraction.

$- \frac{12}{x^{3}} - 12 x^{- 4}$

Step 4.2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{12}{x^{3}} - 12 \frac{1}{x^{4}}$

Step 4.2.5

Combine $- 12$ and $\frac{1}{x^{4}}$ .

$- \frac{12}{x^{3}} + \frac{- 12}{x^{4}}$

Step 4.2.6

Move the negative in front of the fraction.

$- \frac{12}{x^{3}} - \frac{12}{x^{4}}$