Calculus Examples

$2000 + 250 x + 4 x^{\frac{1}{2}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $2000 + 250 x + 4 x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [2000] + \frac{d}{d x} [250 x] + \frac{d}{d x} [4 x^{\frac{1}{2}}]$ .

$\frac{d}{d x} [2000] + \frac{d}{d x} [250 x] + \frac{d}{d x} [4 x^{\frac{1}{2}}]$

Step 1.2

Since $2000$ is constant with respect to $x$ , the derivative of $2000$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [250 x] + \frac{d}{d x} [4 x^{\frac{1}{2}}]$

Step 2

Evaluate $\frac{d}{d x} [250 x]$ .

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Step 2.1

Since $250$ is constant with respect to $x$ , the derivative of $250 x$ with respect to $x$ is $250 \frac{d}{d x} [x]$ .

$0 + 250 \frac{d}{d x} [x] + \frac{d}{d x} [4 x^{\frac{1}{2}}]$

Step 2.2

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

$0 + 250 \cdot 1 + \frac{d}{d x} [4 x^{\frac{1}{2}}]$

Step 2.3

Multiply $250$ by $1$ .

$0 + 250 + \frac{d}{d x} [4 x^{\frac{1}{2}}]$

Step 3

Evaluate $\frac{d}{d x} [4 x^{\frac{1}{2}}]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$0 + 250 + 4 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.2

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

$0 + 250 + 4 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.3

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

$0 + 250 + 4 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.4

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

$0 + 250 + 4 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.5

Combine the numerators over the common denominator.

$0 + 250 + 4 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$0 + 250 + 4 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.6.2

Subtract $2$ from $1$ .

$0 + 250 + 4 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.7

Move the negative in front of the fraction.

$0 + 250 + 4 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.8

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

$0 + 250 + 4 \frac{x^{- \frac{1}{2}}}{2}$

Step 3.9

Combine $4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$0 + 250 + \frac{4 x^{- \frac{1}{2}}}{2}$

Step 3.10

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

$0 + 250 + \frac{4}{2 x^{\frac{1}{2}}}$

Step 3.11

Factor $2$ out of $4$ .

$0 + 250 + \frac{2 \cdot 2}{2 x^{\frac{1}{2}}}$

Step 3.12

Cancel the common factors.

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Step 3.12.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$0 + 250 + \frac{2 \cdot 2}{2 (x^{\frac{1}{2}})}$

Step 3.12.2

Cancel the common factor.

$0 + 250 + \frac{2 \cdot 2}{2 x^{\frac{1}{2}}}$

Step 3.12.3

Rewrite the expression.

$0 + 250 + \frac{2}{x^{\frac{1}{2}}}$

Step 4

Add $0$ and $250$ .

$250 + \frac{2}{x^{\frac{1}{2}}}$