Calculus Examples

$x \sqrt{x}$

Step 1

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 $g (x) = \sqrt{x}$ .

$x \frac{d}{d x} [\sqrt{x}] + \sqrt{x} \frac{d}{d x} [x]$

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

$x \frac{d}{d x} [x^{\frac{1}{2}}] + \sqrt{x} \frac{d}{d x} [x]$

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Simplify.

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Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.2

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

$x \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \frac{d}{d x} [x]$

Step 9

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

$x \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 1$

Step 10

Combine terms.

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Step 10.1

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

$\frac{x}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 1$

Step 10.2

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

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

Step 10.3

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

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Step 10.3.1

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

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Step 10.3.1.1

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

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

Step 10.3.1.2

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

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

Step 10.3.2

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

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

Step 10.3.3

Combine the numerators over the common denominator.

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

Step 10.3.4

Subtract $1$ from $2$ .

$\frac{x^{\frac{1}{2}}}{2} + \sqrt{x} \cdot 1$

Step 10.4

Multiply $\sqrt{x}$ by $1$ .

$\frac{x^{\frac{1}{2}}}{2} + \sqrt{x}$