Calculus Examples

$\frac{d}{d x} (- 4 x^{- 2} - 5 x^{- \frac{5}{2}})$

Step 1

By the Sum Rule, the derivative of $- 4 x^{- 2} - 5 x^{- \frac{5}{2}}$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{- 2}] + \frac{d}{d x} [- 5 x^{- \frac{5}{2}}]$ .

$\frac{d}{d x} [- 4 x^{- 2}] + \frac{d}{d x} [- 5 x^{- \frac{5}{2}}]$

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

$- 4 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [- 5 x^{- \frac{5}{2}}]$

Step 2.2

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

$- 4 (- 2 x^{- 3}) + \frac{d}{d x} [- 5 x^{- \frac{5}{2}}]$

Step 2.3

Multiply $- 2$ by $- 4$ .

$8 x^{- 3} + \frac{d}{d x} [- 5 x^{- \frac{5}{2}}]$

Step 3

Evaluate $\frac{d}{d x} [- 5 x^{- \frac{5}{2}}]$ .

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Step 3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{- \frac{5}{2}}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{- \frac{5}{2}}]$ .

$8 x^{- 3} - 5 \frac{d}{d x} [x^{- \frac{5}{2}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{5}{2}$ .

$8 x^{- 3} - 5 (- \frac{5}{2} x^{- \frac{5}{2} - 1})$

Step 3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 x^{- 3} - 5 (- \frac{5}{2} x^{- \frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 3.4

Combine $- 1$ and $\frac{2}{2}$ .

$8 x^{- 3} - 5 (- \frac{5}{2} x^{- \frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.5

Combine the numerators over the common denominator.

$8 x^{- 3} - 5 (- \frac{5}{2} x^{\frac{- 5 - 1 \cdot 2}{2}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$8 x^{- 3} - 5 (- \frac{5}{2} x^{\frac{- 5 - 2}{2}})$

Step 3.6.2

Subtract $2$ from $- 5$ .

$8 x^{- 3} - 5 (- \frac{5}{2} x^{\frac{- 7}{2}})$

Step 3.7

Move the negative in front of the fraction.

$8 x^{- 3} - 5 (- \frac{5}{2} x^{- \frac{7}{2}})$

Step 3.8

Combine $x^{- \frac{7}{2}}$ and $\frac{5}{2}$ .

$8 x^{- 3} - 5 (- \frac{x^{- \frac{7}{2}} \cdot 5}{2})$

Step 3.9

Multiply $- 1$ by $- 5$ .

$8 x^{- 3} + 5 \frac{x^{- \frac{7}{2}} \cdot 5}{2}$

Step 3.10

Combine $5$ and $\frac{x^{- \frac{7}{2}} \cdot 5}{2}$ .

$8 x^{- 3} + \frac{5 (x^{- \frac{7}{2}} \cdot 5)}{2}$

Step 3.11

Multiply $5$ by $5$ .

$8 x^{- 3} + \frac{25 x^{- \frac{7}{2}}}{2}$

Step 3.12

Move $x^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 x^{- 3} + \frac{25}{2 x^{\frac{7}{2}}}$

Step 4

Simplify each term.

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Step 4.1

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

$8 \frac{1}{x^{3}} + \frac{25}{2 x^{\frac{7}{2}}}$

Step 4.2

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

$\frac{8}{x^{3}} + \frac{25}{2 x^{\frac{7}{2}}}$