Calculus Examples

$f (x) = \cos (e^{x} + \frac{1}{\sqrt{x}})$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

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

Step 5.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.2.2

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

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

Step 5.2.3

Move the negative in front of the fraction.

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $- 1$ .

$- \sin (e^{x} + \frac{1}{x^{\frac{1}{2}}}) (e^{x} - \frac{1}{2} x^{\frac{- 3}{2}})$

Step 10

Move the negative in front of the fraction.

$- \sin (e^{x} + \frac{1}{x^{\frac{1}{2}}}) (e^{x} - \frac{1}{2} x^{- \frac{3}{2}})$

Step 11

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$- \sin (e^{x} + \frac{1}{x^{\frac{1}{2}}}) (e^{x} - \frac{x^{- \frac{3}{2}}}{2})$

Step 12

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \sin (e^{x} + \frac{1}{x^{\frac{1}{2}}}) (e^{x} - \frac{1}{2 x^{\frac{3}{2}}})$