Calculus Examples

$1 + \sqrt{2 x}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\sqrt{2 x}]$

Step 2

Evaluate $\frac{d}{d x} [\sqrt{2 x}]$ .

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Step 2.1

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

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

Step 2.2

Factor $2$ out of $2 x$ .

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

Step 2.3

Apply the product rule to $2 (x)$ .

$0 + \frac{d}{d x} [2^{\frac{1}{2}} x^{\frac{1}{2}}]$

Step 2.4

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

$0 + 2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

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

Step 2.9.2

Subtract $2$ from $1$ .

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

Step 2.10

Move the negative in front of the fraction.

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

Step 2.11

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

$0 + 2^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2}$

Step 2.12

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

$0 + \frac{2^{\frac{1}{2}} x^{- \frac{1}{2}}}{2}$

Step 2.13

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

$0 + \frac{x^{- \frac{1}{2}}}{2 \cdot 2^{- \frac{1}{2}}}$

Step 2.14

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

$0 + \frac{1}{2 \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.15

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

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Step 2.15.1

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

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Step 2.15.1.1

Raise $2$ to the power of $1$ .

$0 + \frac{1}{2^{1} \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.15.1.2

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

$0 + \frac{1}{2^{1 - \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.15.2

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

$0 + \frac{1}{2^{\frac{2}{2} - \frac{1}{2}} x^{\frac{1}{2}}}$

Step 2.15.3

Combine the numerators over the common denominator.

$0 + \frac{1}{2^{\frac{2 - 1}{2}} x^{\frac{1}{2}}}$

Step 2.15.4

Subtract $1$ from $2$ .

$0 + \frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$

Step 3

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

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