Calculus Examples

$f (x) = (6 \sqrt{x} - 2) (5 \sqrt{x} + 7)$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\frac{d}{d x} [(6 x^{\frac{1}{2}} - 2) (5 \sqrt{x} + 7)]$

Step 1.2

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

$\frac{d}{d x} [(6 x^{\frac{1}{2}} - 2) (5 x^{\frac{1}{2}} + 7)]$

Step 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 $f (x) = 6 x^{\frac{1}{2}} - 2$ and $g (x) = 5 x^{\frac{1}{2}} + 7$ .

$(6 x^{\frac{1}{2}} - 2) \frac{d}{d x} [5 x^{\frac{1}{2}} + 7] + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $5 x^{\frac{1}{2}} + 7$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{1}{2}}] + \frac{d}{d x} [7]$ .

$(6 x^{\frac{1}{2}} - 2) (\frac{d}{d x} [5 x^{\frac{1}{2}}] + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 3.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{1}{2}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$(6 x^{\frac{1}{2}} - 2) (5 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 3.3

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

$(6 x^{\frac{1}{2}} - 2) (5 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 4

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

$(6 x^{\frac{1}{2}} - 2) (5 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 5

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

$(6 x^{\frac{1}{2}} - 2) (5 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 6

Combine the numerators over the common denominator.

$(6 x^{\frac{1}{2}} - 2) (5 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$(6 x^{\frac{1}{2}} - 2) (5 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 7.2

Subtract $2$ from $1$ .

$(6 x^{\frac{1}{2}} - 2) (5 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$(6 x^{\frac{1}{2}} - 2) (5 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 8.2

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

$(6 x^{\frac{1}{2}} - 2) (5 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 8.3

Combine $5$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$(6 x^{\frac{1}{2}} - 2) (\frac{5 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 8.4

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

$(6 x^{\frac{1}{2}} - 2) (\frac{5}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [7]) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 9

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$(6 x^{\frac{1}{2}} - 2) (\frac{5}{2 x^{\frac{1}{2}}} + 0) + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 10

Add $\frac{5}{2 x^{\frac{1}{2}}}$ and $0$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) \frac{d}{d x} [6 x^{\frac{1}{2}} - 2]$

Step 11

By the Sum Rule, the derivative of $6 x^{\frac{1}{2}} - 2$ with respect to $x$ is $\frac{d}{d x} [6 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{d}{d x} [6 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2])$

Step 12

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{\frac{1}{2}}$ with respect to $x$ is $6 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2])$

Step 13

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

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 2])$

Step 14

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

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 2])$

Step 15

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

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2])$

Step 16

Combine the numerators over the common denominator.

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2])$

Step 17

Simplify the numerator.

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Step 17.1

Multiply $- 1$ by $2$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 2])$

Step 17.2

Subtract $2$ from $1$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 2])$

Step 18

Move the negative in front of the fraction.

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 2])$

Step 19

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

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (6 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2])$

Step 20

Combine $6$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{6 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2])$

Step 21

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

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{6}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 2])$

Step 22

Factor $2$ out of $6$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{2 \cdot 3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 2])$

Step 23

Cancel the common factors.

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Step 23.1

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

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{2 \cdot 3}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [- 2])$

Step 23.2

Cancel the common factor.

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{2 \cdot 3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 2])$

Step 23.3

Rewrite the expression.

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{3}{x^{\frac{1}{2}}} + \frac{d}{d x} [- 2])$

Step 24

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) (\frac{3}{x^{\frac{1}{2}}} + 0)$

Step 25

Add $\frac{3}{x^{\frac{1}{2}}}$ and $0$ .

$(6 x^{\frac{1}{2}} - 2) \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) \frac{3}{x^{\frac{1}{2}}}$

Step 26

Simplify.

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Step 26.1

Apply the distributive property.

$6 x^{\frac{1}{2}} \frac{5}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + (5 x^{\frac{1}{2}} + 7) \frac{3}{x^{\frac{1}{2}}}$

Step 26.2

Apply the distributive property.

$6 x^{\frac{1}{2}} \frac{5}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3

Combine terms.

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Step 26.3.1

Combine $6$ and $\frac{5}{2 x^{\frac{1}{2}}}$ .

$x^{\frac{1}{2}} \frac{6 \cdot 5}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.2

Multiply $6$ by $5$ .

$x^{\frac{1}{2}} \frac{30}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.3

Combine $x^{\frac{1}{2}}$ and $\frac{30}{2 x^{\frac{1}{2}}}$ .

$\frac{x^{\frac{1}{2}} \cdot 30}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.4

Move $30$ to the left of $x^{\frac{1}{2}}$ .

$\frac{30 \cdot x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.5

Factor $2$ out of $30 x^{\frac{1}{2}}$ .

$\frac{2 (15 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.6

Cancel the common factors.

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Step 26.3.6.1

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

$\frac{2 (15 x^{\frac{1}{2}})}{2 (x^{\frac{1}{2}})} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.6.2

Cancel the common factor.

$\frac{2 (15 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.6.3

Rewrite the expression.

$\frac{15 x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.7

Cancel the common factor.

$\frac{15 x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.8

Divide $15$ by $1$ .

$15 - 2 \frac{5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.9

Combine $- 2$ and $\frac{5}{2 x^{\frac{1}{2}}}$ .

$15 + \frac{- 2 \cdot 5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.10

Multiply $- 2$ by $5$ .

$15 + \frac{- 10}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.11

Factor $2$ out of $- 10$ .

$15 + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.12

Cancel the common factors.

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Step 26.3.12.1

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

$15 + \frac{2 \cdot - 5}{2 (x^{\frac{1}{2}})} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.12.2

Cancel the common factor.

$15 + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.12.3

Rewrite the expression.

$15 + \frac{- 5}{x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.13

Move the negative in front of the fraction.

$15 - \frac{5}{x^{\frac{1}{2}}} + 5 x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.14

Combine $5$ and $\frac{3}{x^{\frac{1}{2}}}$ .

$15 - \frac{5}{x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{5 \cdot 3}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.15

Multiply $5$ by $3$ .

$15 - \frac{5}{x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{15}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.16

Combine $x^{\frac{1}{2}}$ and $\frac{15}{x^{\frac{1}{2}}}$ .

$15 - \frac{5}{x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}} \cdot 15}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.17

Move $15$ to the left of $x^{\frac{1}{2}}$ .

$15 - \frac{5}{x^{\frac{1}{2}}} + \frac{15 \cdot x^{\frac{1}{2}}}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.18

Cancel the common factor.

$15 - \frac{5}{x^{\frac{1}{2}}} + \frac{15 x^{\frac{1}{2}}}{x^{\frac{1}{2}}} + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.19

Divide $15$ by $1$ .

$15 - \frac{5}{x^{\frac{1}{2}}} + 15 + 7 \frac{3}{x^{\frac{1}{2}}}$

Step 26.3.20

Combine $7$ and $\frac{3}{x^{\frac{1}{2}}}$ .

$15 - \frac{5}{x^{\frac{1}{2}}} + 15 + \frac{7 \cdot 3}{x^{\frac{1}{2}}}$

Step 26.3.21

Multiply $7$ by $3$ .

$15 - \frac{5}{x^{\frac{1}{2}}} + 15 + \frac{21}{x^{\frac{1}{2}}}$

Step 26.3.22

Add $15$ and $15$ .

$30 - \frac{5}{x^{\frac{1}{2}}} + \frac{21}{x^{\frac{1}{2}}}$

Step 26.3.23

Combine the numerators over the common denominator.

$30 + \frac{- 5 + 21}{x^{\frac{1}{2}}}$

Step 26.3.24

Add $- 5$ and $21$ .

$30 + \frac{16}{x^{\frac{1}{2}}}$