Calculus Examples

7 e^{x} \sqrt{x}

$7 e^{x} \sqrt{x}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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} [7 e^{x} x^{\frac{1}{2}}]

$\frac{d}{d x} [7 e^{x} x^{\frac{1}{2}}]$

Step 1.2

Since

7

$7$ is constant with respect to

x

$x$ , the derivative of

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

$7 e^{x} x^{\frac{1}{2}}$ with respect to

x

$x$ is

7 \frac{d}{d x} [e^{x} x^{\frac{1}{2}}]

$7 \frac{d}{d x} [e^{x} x^{\frac{1}{2}}]$ .

7 \frac{d}{d x} [e^{x} x^{\frac{1}{2}}]

$7 \frac{d}{d x} [e^{x} x^{\frac{1}{2}}]$

7 \frac{d}{d x} [e^{x} x^{\frac{1}{2}}]

$7 \frac{d}{d x} [e^{x} x^{\frac{1}{2}}]$

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) = e^{x}

$f (x) = e^{x}$ and

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

$g (x) = x^{\frac{1}{2}}$ .

7 (e^{x} \frac{d}{d x} [x^{\frac{1}{2}}] + x^{\frac{1}{2}} \frac{d}{d x} [e^{x}])

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

Step 3

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}

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

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

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

Step 4

To write

- 1

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

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 5

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 6

Combine the numerators over the common denominator.

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

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 7.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

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

Step 8.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

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

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

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

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

Step 8.3

Combine

e^{x}

$e^{x}$ and

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

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

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

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

Step 8.4

Move

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

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

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

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

7 (\frac{e^{x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{d}{d x} [e^{x}])

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

7 (\frac{e^{x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{d}{d x} [e^{x}])

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

Step 9

Differentiate using the Exponential Rule which states that

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

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

a^{x} \ln (a)

$a^{x} \ln (a)$ where

a

$a$ =

e

$e$ .

7 (\frac{e^{x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} e^{x})

$7 (\frac{e^{x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} e^{x})$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

7 \frac{e^{x}}{2 x^{\frac{1}{2}}} + 7 (x^{\frac{1}{2}} e^{x})

$7 \frac{e^{x}}{2 x^{\frac{1}{2}}} + 7 (x^{\frac{1}{2}} e^{x})$

Step 10.2

Combine

7

$7$ and

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

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

\frac{7 e^{x}}{2 x^{\frac{1}{2}}} + 7 x^{\frac{1}{2}} e^{x}

$\frac{7 e^{x}}{2 x^{\frac{1}{2}}} + 7 x^{\frac{1}{2}} e^{x}$

Step 10.3

Reorder terms.

7 e^{x} x^{\frac{1}{2}} + \frac{7 e^{x}}{2 x^{\frac{1}{2}}}

$7 e^{x} x^{\frac{1}{2}} + \frac{7 e^{x}}{2 x^{\frac{1}{2}}}$

7 e^{x} x^{\frac{1}{2}} + \frac{7 e^{x}}{2 x^{\frac{1}{2}}}

$7 e^{x} x^{\frac{1}{2}} + \frac{7 e^{x}}{2 x^{\frac{1}{2}}}$