Calculus Examples

$80 - 0.08 x - 550 x^{- 1}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $80 - 0.08 x - 550 x^{- 1}$ with respect to $x$ is $\frac{d}{d x} [80] + \frac{d}{d x} [- 0.08 x] + \frac{d}{d x} [- 550 x^{- 1}]$ .

$\frac{d}{d x} [80] + \frac{d}{d x} [- 0.08 x] + \frac{d}{d x} [- 550 x^{- 1}]$

Step 1.2

Since $80$ is constant with respect to $x$ , the derivative of $80$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 0.08 x] + \frac{d}{d x} [- 550 x^{- 1}]$

Step 2

Evaluate $\frac{d}{d x} [- 0.08 x]$ .

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Step 2.1

Since $- 0.08$ is constant with respect to $x$ , the derivative of $- 0.08 x$ with respect to $x$ is $- 0.08 \frac{d}{d x} [x]$ .

$0 - 0.08 \frac{d}{d x} [x] + \frac{d}{d x} [- 550 x^{- 1}]$

Step 2.2

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

$0 - 0.08 \cdot 1 + \frac{d}{d x} [- 550 x^{- 1}]$

Step 2.3

Multiply $- 0.08$ by $1$ .

$0 - 0.08 + \frac{d}{d x} [- 550 x^{- 1}]$

Step 3

Evaluate $\frac{d}{d x} [- 550 x^{- 1}]$ .

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Step 3.1

Since $- 550$ is constant with respect to $x$ , the derivative of $- 550 x^{- 1}$ with respect to $x$ is $- 550 \frac{d}{d x} [x^{- 1}]$ .

$0 - 0.08 - 550 \frac{d}{d x} [x^{- 1}]$

Step 3.2

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

$0 - 0.08 - 550 (- x^{- 2})$

Step 3.3

Multiply $- 1$ by $- 550$ .

$0 - 0.08 + 550 x^{- 2}$

Step 4

Simplify.

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Step 4.1

Subtract $0.08$ from $0$ .

$- 0.08 + 550 x^{- 2}$

Step 4.2

Reorder terms.

$550 x^{- 2} - 0.08$

Step 4.3

Simplify each term.

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Step 4.3.1

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

$550 \frac{1}{x^{2}} - 0.08$

Step 4.3.2

Combine $550$ and $\frac{1}{x^{2}}$ .

$\frac{550}{x^{2}} - 0.08$