Calculus Examples

$f (x) = x (50 - 0.1 \sqrt{x}) - (35 x + 500)$

Step 1

By the Sum Rule, the derivative of $x (50 - 0.1 \sqrt{x}) - (35 x + 500)$ with respect to $x$ is $\frac{d}{d x} [x (50 - 0.1 \sqrt{x})] + \frac{d}{d x} [- (35 x + 500)]$ .

$\frac{d}{d x} [x (50 - 0.1 \sqrt{x})] + \frac{d}{d x} [- (35 x + 500)]$

Step 2

Evaluate $\frac{d}{d x} [x (50 - 0.1 \sqrt{x})]$ .

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Step 2.1

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

$\frac{d}{d x} [x (50 - 0.1 x^{\frac{1}{2}})] + \frac{d}{d x} [- (35 x + 500)]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 50 - 0.1 x^{\frac{1}{2}}$ .

$x \frac{d}{d x} [50 - 0.1 x^{\frac{1}{2}}] + (50 - 0.1 x^{\frac{1}{2}}) \frac{d}{d x} [x] + \frac{d}{d x} [- (35 x + 500)]$

Step 2.3

By the Sum Rule, the derivative of $50 - 0.1 x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [50] + \frac{d}{d x} [- 0.1 x^{\frac{1}{2}}]$ .

$x (\frac{d}{d x} [50] + \frac{d}{d x} [- 0.1 x^{\frac{1}{2}}]) + (50 - 0.1 x^{\frac{1}{2}}) \frac{d}{d x} [x] + \frac{d}{d x} [- (35 x + 500)]$

Step 2.4

Since $50$ is constant with respect to $x$ , the derivative of $50$ with respect to $x$ is $0$ .

$x (0 + \frac{d}{d x} [- 0.1 x^{\frac{1}{2}}]) + (50 - 0.1 x^{\frac{1}{2}}) \frac{d}{d x} [x] + \frac{d}{d x} [- (35 x + 500)]$

Step 2.5

Since $- 0.1$ is constant with respect to $x$ , the derivative of $- 0.1 x^{\frac{1}{2}}$ with respect to $x$ is $- 0.1 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$x (0 - 0.1 \frac{d}{d x} [x^{\frac{1}{2}}]) + (50 - 0.1 x^{\frac{1}{2}}) \frac{d}{d x} [x] + \frac{d}{d x} [- (35 x + 500)]$

Step 2.6

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

$x (0 - 0.1 (\frac{1}{2} x^{\frac{1}{2} - 1})) + (50 - 0.1 x^{\frac{1}{2}}) \frac{d}{d x} [x] + \frac{d}{d x} [- (35 x + 500)]$

Step 2.7

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

$x (0 - 0.1 (\frac{1}{2} x^{\frac{1}{2} - 1})) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.8

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

$x (0 - 0.1 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.9

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

$x (0 - 0.1 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.10

Combine the numerators over the common denominator.

$x (0 - 0.1 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Multiply $- 1$ by $2$ .

$x (0 - 0.1 (\frac{1}{2} x^{\frac{1 - 2}{2}})) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.11.2

Subtract $2$ from $1$ .

$x (0 - 0.1 (\frac{1}{2} x^{\frac{- 1}{2}})) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.12

Move the negative in front of the fraction.

$x (0 - 0.1 (\frac{1}{2} x^{- \frac{1}{2}})) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.13

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

$x (0 - 0.1 \frac{x^{- \frac{1}{2}}}{2}) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.14

Combine $- 0.1$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$x (0 + \frac{- 0.1 x^{- \frac{1}{2}}}{2}) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.15

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

$x (0 + \frac{- 0.1}{2 x^{\frac{1}{2}}}) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.16

Move the negative in front of the fraction.

$x (0 - \frac{0.1}{2 x^{\frac{1}{2}}}) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.17

Subtract $\frac{0.1}{2 x^{\frac{1}{2}}}$ from $0$ .

$x (- \frac{0.1}{2 x^{\frac{1}{2}}}) + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.18

Combine $x$ and $\frac{0.1}{2 x^{\frac{1}{2}}}$ .

$- \frac{x \cdot 0.1}{2 x^{\frac{1}{2}}} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.19

Move $0.1$ to the left of $x$ .

$- \frac{0.1 \cdot x}{2 x^{\frac{1}{2}}} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.20

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

$- \frac{0.1 x \cdot x^{- \frac{1}{2}}}{2} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.21

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.21.1

Move $x^{- \frac{1}{2}}$ .

$- \frac{0.1 (x^{- \frac{1}{2}} x)}{2} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.21.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 2.21.2.1

Raise $x$ to the power of $1$ .

$- \frac{0.1 (x^{- \frac{1}{2}} x^{1})}{2} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.21.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{0.1 x^{- \frac{1}{2} + 1}}{2} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.21.3

Write $1$ as a fraction with a common denominator.

$- \frac{0.1 x^{- \frac{1}{2} + \frac{2}{2}}}{2} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.21.4

Combine the numerators over the common denominator.

$- \frac{0.1 x^{\frac{- 1 + 2}{2}}}{2} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.21.5

Add $- 1$ and $2$ .

$- \frac{0.1 x^{\frac{1}{2}}}{2} + (50 - 0.1 x^{\frac{1}{2}}) \cdot 1 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.22

Multiply $50 - 0.1 x^{\frac{1}{2}}$ by $1$ .

$- \frac{0.1 x^{\frac{1}{2}}}{2} + 50 - 0.1 x^{\frac{1}{2}} + \frac{d}{d x} [- (35 x + 500)]$

Step 2.23

To write $- 0.1 x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{0.1 x^{\frac{1}{2}}}{2} - 0.1 x^{\frac{1}{2}} \cdot \frac{2}{2} + 50 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.24

Combine $- 0.1 x^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$- \frac{0.1 x^{\frac{1}{2}}}{2} + \frac{- 0.1 x^{\frac{1}{2}} \cdot 2}{2} + 50 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.25

Combine the numerators over the common denominator.

$\frac{- 0.1 x^{\frac{1}{2}} - 0.1 x^{\frac{1}{2}} \cdot 2}{2} + 50 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.26

Multiply $2$ by $- 0.1$ .

$\frac{- 0.1 x^{\frac{1}{2}} - 0.2 x^{\frac{1}{2}}}{2} + 50 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.27

Subtract $0.2 x^{\frac{1}{2}}$ from $- 0.1 x^{\frac{1}{2}}$ .

$\frac{- 0.3 x^{\frac{1}{2}}}{2} + 50 + \frac{d}{d x} [- (35 x + 500)]$

Step 2.28

Move the negative in front of the fraction.

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 + \frac{d}{d x} [- (35 x + 500)]$

Step 3

Evaluate $\frac{d}{d x} [- (35 x + 500)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (35 x + 500)$ with respect to $x$ is $- \frac{d}{d x} [35 x + 500]$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - \frac{d}{d x} [35 x + 500]$

Step 3.2

By the Sum Rule, the derivative of $35 x + 500$ with respect to $x$ is $\frac{d}{d x} [35 x] + \frac{d}{d x} [500]$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - (\frac{d}{d x} [35 x] + \frac{d}{d x} [500])$

Step 3.3

Since $35$ is constant with respect to $x$ , the derivative of $35 x$ with respect to $x$ is $35 \frac{d}{d x} [x]$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - (35 \frac{d}{d x} [x] + \frac{d}{d x} [500])$

Step 3.4

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

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - (35 \cdot 1 + \frac{d}{d x} [500])$

Step 3.5

Since $500$ is constant with respect to $x$ , the derivative of $500$ with respect to $x$ is $0$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - (35 \cdot 1 + 0)$

Step 3.6

Multiply $35$ by $1$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - (35 + 0)$

Step 3.7

Add $35$ and $0$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - 1 \cdot 35$

Step 3.8

Multiply $- 1$ by $35$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 50 - 35$

Step 4

Simplify.

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Step 4.1

Subtract $35$ from $50$ .

$- \frac{0.3 x^{\frac{1}{2}}}{2} + 15$

Step 4.2

Reorder terms.

$15 - \frac{0.3 x^{\frac{1}{2}}}{2}$

Step 4.3

Simplify each term.

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Step 4.3.1

Factor $0.3$ out of $0.3 x^{\frac{1}{2}}$ .

$15 - \frac{0.3 (x^{\frac{1}{2}})}{2}$

Step 4.3.2

Factor $2$ out of $2$ .

$15 - \frac{0.3 (x^{\frac{1}{2}})}{2 (1)}$

Step 4.3.3

Separate fractions.

$15 - (\frac{0.3}{2} \cdot \frac{x^{\frac{1}{2}}}{1})$

Step 4.3.4

Divide $0.3$ by $2$ .

$15 - (0.15 \frac{x^{\frac{1}{2}}}{1})$

Step 4.3.5

Divide $x^{\frac{1}{2}}$ by $1$ .

$15 - (0.15 x^{\frac{1}{2}})$

Step 4.3.6

Multiply $0.15$ by $- 1$ .

$15 - 0.15 x^{\frac{1}{2}}$