Calculus Examples

$\ln (2 x^{0.5})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 2 x^{0.5}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2 x^{0.5}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 x^{0.5}]$

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [2 x^{0.5}]$

Step 1.3

Replace all occurrences of $u$ with $2 x^{0.5}$ .

$\frac{1}{2 x^{0.5}} \frac{d}{d x} [2 x^{0.5}]$

Step 2

Differentiate.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{0.5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{0.5}]$ .

$\frac{1}{2 x^{0.5}} (2 \frac{d}{d x} [x^{0.5}])$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $2$ and $\frac{1}{2 x^{0.5}}$ .

$\frac{2}{2 x^{0.5}} \frac{d}{d x} [x^{0.5}]$

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{2}{2 x^{0.5}} \frac{d}{d x} [x^{0.5}]$

Step 2.2.2.2

Rewrite the expression.

$\frac{1}{x^{0.5}} \frac{d}{d x} [x^{0.5}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.5$ .

$\frac{1}{x^{0.5}} (0.5 x^{- 0.5})$

Step 2.4

Combine fractions.

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Step 2.4.1

Combine $0.5$ and $\frac{1}{x^{0.5}}$ .

$\frac{0.5}{x^{0.5}} x^{- 0.5}$

Step 2.4.2

Combine $\frac{0.5}{x^{0.5}}$ and $x^{- 0.5}$ .

$\frac{0.5 x^{- 0.5}}{x^{0.5}}$

Step 2.4.3

Move $x^{- 0.5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{0.5}{x^{0.5} x^{0.5}}$

Step 3

Simplify the denominator.

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Step 3.1

Multiply $x^{0.5}$ by $x^{0.5}$ by adding the exponents.

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Step 3.1.1

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

$\frac{0.5}{x^{0.5 + 0.5}}$

Step 3.1.2

Add $0.5$ and $0.5$ .

$\frac{0.5}{x^{1}}$

Step 3.2

Simplify $x^{1}$ .

$\frac{0.5}{x}$