Calculus Examples

$f (x) = x \cos (\sqrt{x})$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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Step 15.1

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

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

Step 15.2

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

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Step 15.2.1

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

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

Step 15.2.2

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

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

Step 15.3

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

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

Step 15.4

Combine the numerators over the common denominator.

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

Step 15.5

Add $- 1$ and $2$ .

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

Step 16

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

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

Step 17

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

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