Calculus Examples

$f (x) = 0.002 x^{3} + 8 + \frac{7626}{x}$

Step 1

By the Sum Rule, the derivative of $0.002 x^{3} + 8 + \frac{7626}{x}$ with respect to $x$ is $\frac{d}{d x} [0.002 x^{3}] + \frac{d}{d x} [8] + \frac{d}{d x} [\frac{7626}{x}]$ .

$\frac{d}{d x} [0.002 x^{3}] + \frac{d}{d x} [8] + \frac{d}{d x} [\frac{7626}{x}]$

Step 2

Evaluate $\frac{d}{d x} [0.002 x^{3}]$ .

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Step 2.1

Since $0.002$ is constant with respect to $x$ , the derivative of $0.002 x^{3}$ with respect to $x$ is $0.002 \frac{d}{d x} [x^{3}]$ .

$0.002 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [8] + \frac{d}{d x} [\frac{7626}{x}]$

Step 2.2

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

$0.002 (3 x^{2}) + \frac{d}{d x} [8] + \frac{d}{d x} [\frac{7626}{x}]$

Step 2.3

Multiply $3$ by $0.002$ .

$0.006 x^{2} + \frac{d}{d x} [8] + \frac{d}{d x} [\frac{7626}{x}]$

Step 3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$0.006 x^{2} + 0 + \frac{d}{d x} [\frac{7626}{x}]$

Step 4

Evaluate $\frac{d}{d x} [\frac{7626}{x}]$ .

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Step 4.1

Since $7626$ is constant with respect to $x$ , the derivative of $\frac{7626}{x}$ with respect to $x$ is $7626 \frac{d}{d x} [\frac{1}{x}]$ .

$0.006 x^{2} + 0 + 7626 \frac{d}{d x} [\frac{1}{x}]$

Step 4.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$0.006 x^{2} + 0 + 7626 \frac{d}{d x} [x^{- 1}]$

Step 4.3

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

$0.006 x^{2} + 0 + 7626 (- x^{- 2})$

Step 4.4

Multiply $- 1$ by $7626$ .

$0.006 x^{2} + 0 - 7626 x^{- 2}$

Step 5

Simplify.

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Step 5.1

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

$0.006 x^{2} + 0 - 7626 \frac{1}{x^{2}}$

Step 5.2

Combine terms.

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Step 5.2.1

Add $0.006 x^{2}$ and $0$ .

$0.006 x^{2} - 7626 \frac{1}{x^{2}}$

Step 5.2.2

Combine $- 7626$ and $\frac{1}{x^{2}}$ .

$0.006 x^{2} + \frac{- 7626}{x^{2}}$

Step 5.2.3

Move the negative in front of the fraction.

$0.006 x^{2} - \frac{7626}{x^{2}}$