Calculus Examples

$y = \sin (2 \sqrt{x})$

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $2 x^{\frac{1}{2}}$ .

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

Step 2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

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

Step 2.3

Replace all occurrences of $u$ with $2 x^{\frac{1}{2}}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

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

$\cos (2 x^{\frac{1}{2}}) \frac{2}{2 x^{\frac{1}{2}}}$

Step 12

Cancel the common factor.

$\cos (2 x^{\frac{1}{2}}) \frac{2}{2 x^{\frac{1}{2}}}$

Step 13

Rewrite the expression.

$\cos (2 x^{\frac{1}{2}}) \frac{1}{x^{\frac{1}{2}}}$

Step 14

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

$\frac{\cos (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$