Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 x^{- 3} - 4 x^{- 2})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Evaluate $\frac{d}{d x} [2 x^{- 3}]$ .

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Step 3.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{- 3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{- 3}]$ .

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $- 3$ by $2$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $- 2$ by $- 4$ .

$- 6 x^{- 4} + 8 x^{- 3}$

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

$8 x^{- 3} - 6 x^{- 4}$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

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

$8 \frac{1}{x^{3}} - 6 x^{- 4}$

Step 3.4.2.2

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

$\frac{8}{x^{3}} - 6 x^{- 4}$

Step 3.4.2.3

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

$\frac{8}{x^{3}} - 6 \frac{1}{x^{4}}$

Step 3.4.2.4

Combine $- 6$ and $\frac{1}{x^{4}}$ .

$\frac{8}{x^{3}} + \frac{- 6}{x^{4}}$

Step 3.4.2.5

Move the negative in front of the fraction.

$\frac{8}{x^{3}} - \frac{6}{x^{4}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{8}{x^{3}} - \frac{6}{x^{4}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{8}{x^{3}} - \frac{6}{x^{4}}$