Calculus Examples

$10000 - 2500 x + 3750 \sqrt{16 + x^{2}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $10000 - 2500 x + 3750 \sqrt{16 + x^{2}}$ with respect to $x$ is $\frac{d}{d x} [10000] + \frac{d}{d x} [- 2500 x] + \frac{d}{d x} [3750 \sqrt{16 + x^{2}}]$ .

$\frac{d}{d x} [10000] + \frac{d}{d x} [- 2500 x] + \frac{d}{d x} [3750 \sqrt{16 + x^{2}}]$

Step 1.2

Since $10000$ is constant with respect to $x$ , the derivative of $10000$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2500 x] + \frac{d}{d x} [3750 \sqrt{16 + x^{2}}]$

Step 2

Evaluate $\frac{d}{d x} [- 2500 x]$ .

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Step 2.1

Since $- 2500$ is constant with respect to $x$ , the derivative of $- 2500 x$ with respect to $x$ is $- 2500 \frac{d}{d x} [x]$ .

$0 - 2500 \frac{d}{d x} [x] + \frac{d}{d x} [3750 \sqrt{16 + x^{2}}]$

Step 2.2

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

$0 - 2500 \cdot 1 + \frac{d}{d x} [3750 \sqrt{16 + x^{2}}]$

Step 2.3

Multiply $- 2500$ by $1$ .

$0 - 2500 + \frac{d}{d x} [3750 \sqrt{16 + x^{2}}]$

Step 3

Evaluate $\frac{d}{d x} [3750 \sqrt{16 + x^{2}}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 + x^{2}}$ as ${(16 + x^{2})}^{\frac{1}{2}}$ .

$0 - 2500 + \frac{d}{d x} [3750 {(16 + x^{2})}^{\frac{1}{2}}]$

Step 3.2

Since $3750$ is constant with respect to $x$ , the derivative of $3750 {(16 + x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $3750 \frac{d}{d x} [{(16 + x^{2})}^{\frac{1}{2}}]$ .

$0 - 2500 + 3750 \frac{d}{d x} [{(16 + x^{2})}^{\frac{1}{2}}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 16 + x^{2}$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $16 + x^{2}$ .

$0 - 2500 + 3750 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [16 + x^{2}])$

Step 3.3.2

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

$0 - 2500 + 3750 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [16 + x^{2}])$

Step 3.3.3

Replace all occurrences of $u$ with $16 + x^{2}$ .

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [16 + x^{2}])$

Step 3.4

By the Sum Rule, the derivative of $16 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [x^{2}]$ .

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1}{2} - 1} (\frac{d}{d x} [16] + \frac{d}{d x} [x^{2}]))$

Step 3.5

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [x^{2}]))$

Step 3.6

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

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1}{2} - 1} (0 + 2 x))$

Step 3.7

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

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 + 2 x))$

Step 3.8

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

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 + 2 x))$

Step 3.9

Combine the numerators over the common denominator.

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1 - 1 \cdot 2}{2}} (0 + 2 x))$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{1 - 2}{2}} (0 + 2 x))$

Step 3.10.2

Subtract $2$ from $1$ .

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{\frac{- 1}{2}} (0 + 2 x))$

Step 3.11

Move the negative in front of the fraction.

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{- \frac{1}{2}} (0 + 2 x))$

Step 3.12

Add $0$ and $2 x$ .

$0 - 2500 + 3750 (\frac{1}{2} {(16 + x^{2})}^{- \frac{1}{2}} (2 x))$

Step 3.13

Combine $\frac{1}{2}$ and ${(16 + x^{2})}^{- \frac{1}{2}}$ .

$0 - 2500 + 3750 (\frac{{(16 + x^{2})}^{- \frac{1}{2}}}{2} (2 x))$

Step 3.14

Combine $2$ and $\frac{{(16 + x^{2})}^{- \frac{1}{2}}}{2}$ .

$0 - 2500 + 3750 (\frac{2 {(16 + x^{2})}^{- \frac{1}{2}}}{2} x)$

Step 3.15

Combine $\frac{2 {(16 + x^{2})}^{- \frac{1}{2}}}{2}$ and $x$ .

$0 - 2500 + 3750 \frac{2 {(16 + x^{2})}^{- \frac{1}{2}} x}{2}$

Step 3.16

Move ${(16 + x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - 2500 + 3750 \frac{2 x}{2 {(16 + x^{2})}^{\frac{1}{2}}}$

Step 3.17

Cancel the common factor.

$0 - 2500 + 3750 \frac{2 x}{2 {(16 + x^{2})}^{\frac{1}{2}}}$

Step 3.18

Rewrite the expression.

$0 - 2500 + 3750 \frac{x}{{(16 + x^{2})}^{\frac{1}{2}}}$

Step 3.19

Combine $3750$ and $\frac{x}{{(16 + x^{2})}^{\frac{1}{2}}}$ .

$0 - 2500 + \frac{3750 x}{{(16 + x^{2})}^{\frac{1}{2}}}$

Step 4

Simplify.

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Step 4.1

Subtract $2500$ from $0$ .

$- 2500 + \frac{3750 x}{{(16 + x^{2})}^{\frac{1}{2}}}$

Step 4.2

Reorder terms.

$\frac{3750 x}{{(16 + x^{2})}^{\frac{1}{2}}} - 2500$