Calculus Examples

\sqrt{x} (x - 10)

$\sqrt{x} (x - 10)$

Step 1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

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

\frac{d}{d x} [x^{\frac{1}{2}} (x - 10)]

$\frac{d}{d x} [x^{\frac{1}{2}} (x - 10)]$

Step 2

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x^{\frac{1}{2}}

$f (x) = x^{\frac{1}{2}}$ and

g (x) = x - 10

$g (x) = x - 10$ .

x^{\frac{1}{2}} \frac{d}{d x} [x - 10] + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]

$x^{\frac{1}{2}} \frac{d}{d x} [x - 10] + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of

x - 10

$x - 10$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 10]

$\frac{d}{d x} [x] + \frac{d}{d x} [- 10]$ .

x^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 10]) + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]

$x^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 10]) + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

x^{\frac{1}{2}} (1 + \frac{d}{d x} [- 10]) + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]

$x^{\frac{1}{2}} (1 + \frac{d}{d x} [- 10]) + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.3

Since

- 10

$- 10$ is constant with respect to

x

$x$ , the derivative of

- 10

$- 10$ with respect to

x

$x$ is

0

$0$ .

x^{\frac{1}{2}} (1 + 0) + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]

$x^{\frac{1}{2}} (1 + 0) + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add

1

$1$ and

0

$0$ .

x^{\frac{1}{2}} \cdot 1 + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]

$x^{\frac{1}{2}} \cdot 1 + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.4.2

Multiply

x^{\frac{1}{2}}

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

1

$1$ .

x^{\frac{1}{2}} + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]

$x^{\frac{1}{2}} + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]$

x^{\frac{1}{2}} + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]

$x^{\frac{1}{2}} + (x - 10) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.5

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} - 1})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} - 1})$

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} - 1})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 4

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 5

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 6

Combine the numerators over the common denominator.

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply

- 1

$- 1$ by

2

$2$ .

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1 - 2}{2}})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 7.2

Subtract

2

$2$ from

1

$1$ .

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{- 1}{2}})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{- 1}{2}})$

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{- 1}{2}})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 8

Move the negative in front of the fraction.

x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{- \frac{1}{2}})

$x^{\frac{1}{2}} + (x - 10) (\frac{1}{2} x^{- \frac{1}{2}})$

Step 9

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x^{- \frac{1}{2}}

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

x^{\frac{1}{2}} + (x - 10) \frac{x^{- \frac{1}{2}}}{2}

$x^{\frac{1}{2}} + (x - 10) \frac{x^{- \frac{1}{2}}}{2}$

Step 10

Move

x^{- \frac{1}{2}}

$x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

x^{\frac{1}{2}} + (x - 10) \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + (x - 10) \frac{1}{2 x^{\frac{1}{2}}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

x^{\frac{1}{2}} + x \frac{1}{2 x^{\frac{1}{2}}} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + x \frac{1}{2 x^{\frac{1}{2}}} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2

Combine terms.

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Step 11.2.1

Combine

x

$x$ and

\frac{1}{2 x^{\frac{1}{2}}}

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

x^{\frac{1}{2}} + \frac{x}{2 x^{\frac{1}{2}}} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x}{2 x^{\frac{1}{2}}} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2.2

Move

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ to the numerator using the negative exponent rule

\frac{1}{b^{n}} = b^{- n}

$\frac{1}{b^{n}} = b^{- n}$ .

x^{\frac{1}{2}} + \frac{x \cdot x^{- \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x \cdot x^{- \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2.3

Multiply

x

$x$ by

x^{- \frac{1}{2}}

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

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Step 11.2.3.1

Multiply

x

$x$ by

x^{- \frac{1}{2}}

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

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Step 11.2.3.1.1

Raise

x

$x$ to the power of

1

$1$ .

x^{\frac{1}{2}} + \frac{x^{1} x^{- \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{1} x^{- \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2.3.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x^{\frac{1}{2}} + \frac{x^{1 - \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{1 - \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

x^{\frac{1}{2}} + \frac{x^{1 - \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{1 - \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2.3.2

Write

1

$1$ as a fraction with a common denominator.

x^{\frac{1}{2}} + \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2.3.3

Combine the numerators over the common denominator.

x^{\frac{1}{2}} + \frac{x^{\frac{2 - 1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{\frac{2 - 1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2.3.4

Subtract

1

$1$ from

2

$2$ .

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - 10 \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.2.4

Combine

- 10

$- 10$ and

\frac{1}{2 x^{\frac{1}{2}}}

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

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{- 10}{2 x^{\frac{1}{2}}}

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{- 10}{2 x^{\frac{1}{2}}}$

Step 11.2.5

Factor

2

$2$ out of

- 10

$- 10$ .

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}}

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

Step 11.2.6

Cancel the common factors.

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Step 11.2.6.1

Factor

2

$2$ out of

2 x^{\frac{1}{2}}

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

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 5}{2 (x^{\frac{1}{2}})}

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

Step 11.2.6.2

Cancel the common factor.

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 5}{2 x^{\frac{1}{2}}}

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

Step 11.2.6.3

Rewrite the expression.

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{- 5}{x^{\frac{1}{2}}}

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

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{- 5}{x^{\frac{1}{2}}}

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

Step 11.2.7

Move the negative in front of the fraction.

x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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

Step 11.2.8

To write

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

x^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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

Step 11.2.9

Combine

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ and

\frac{2}{2}

$\frac{2}{2}$ .

\frac{x^{\frac{1}{2}} \cdot 2}{2} + \frac{x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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

Step 11.2.10

Combine the numerators over the common denominator.

\frac{x^{\frac{1}{2}} \cdot 2 + x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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

Step 11.2.11

Move

2

$2$ to the left of

x^{\frac{1}{2}}

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

\frac{2 \cdot x^{\frac{1}{2}} + x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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

Step 11.2.12

Add

2 x^{\frac{1}{2}}

$2 x^{\frac{1}{2}}$ and

x^{\frac{1}{2}}

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

\frac{3 x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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

\frac{3 x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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

\frac{3 x^{\frac{1}{2}}}{2} - \frac{5}{x^{\frac{1}{2}}}

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