Calculus Examples

$y = 4 x^{- 2} - 3 x^{- 1}$

Step 1

By the Sum Rule, the derivative of $4 x^{- 2} - 3 x^{- 1}$ with respect to $x$ is $\frac{d}{d x} [4 x^{- 2}] + \frac{d}{d x} [- 3 x^{- 1}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [4 x^{- 2}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{- 2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{- 2}]$ .

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

Step 2.2

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

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

Step 2.3

Multiply $- 2$ by $4$ .

$- 8 x^{- 3} + \frac{d}{d x} [- 3 x^{- 1}]$

Step 3

Evaluate $\frac{d}{d x} [- 3 x^{- 1}]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{- 1}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{- 1}]$ .

$- 8 x^{- 3} - 3 \frac{d}{d x} [x^{- 1}]$

Step 3.2

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

$- 8 x^{- 3} - 3 (- x^{- 2})$

Step 3.3

Multiply $- 1$ by $- 3$ .

$- 8 x^{- 3} + 3 x^{- 2}$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$3 x^{- 2} - 8 x^{- 3}$

Step 4.2

Simplify each term.

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Step 4.2.1

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

$3 \frac{1}{x^{2}} - 8 x^{- 3}$

Step 4.2.2

Combine $3$ and $\frac{1}{x^{2}}$ .

$\frac{3}{x^{2}} - 8 x^{- 3}$

Step 4.2.3

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

$\frac{3}{x^{2}} - 8 \frac{1}{x^{3}}$

Step 4.2.4

Combine $- 8$ and $\frac{1}{x^{3}}$ .

$\frac{3}{x^{2}} + \frac{- 8}{x^{3}}$

Step 4.2.5

Move the negative in front of the fraction.

$\frac{3}{x^{2}} - \frac{8}{x^{3}}$