Calculus Examples

$y = \log (\sqrt{x}) + \tan (x^{e})$

Step 1

By the Sum Rule, the derivative of $\log (\sqrt{x}) + \tan (x^{e})$ with respect to $x$ is $\frac{d}{d x} [\log (\sqrt{x})] + \frac{d}{d x} [\tan (x^{e})]$ .

$\frac{d}{d x} [\log (\sqrt{x})] + \frac{d}{d x} [\tan (x^{e})]$

Step 2

Evaluate $\frac{d}{d x} [\log (\sqrt{x})]$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

The derivative of $\log (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \ln (10)}$ .

$\frac{1}{u_{1} \ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\tan (x^{e})]$

Step 2.2.3

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

$\frac{1}{x^{\frac{1}{2}} \ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\tan (x^{e})]$

Step 2.3

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

$\frac{1}{x^{\frac{1}{2}} \ln (10)} (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [\tan (x^{e})]$

Step 2.4

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

$\frac{1}{x^{\frac{1}{2}} \ln (10)} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [\tan (x^{e})]$

Step 2.5

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

$\frac{1}{x^{\frac{1}{2}} \ln (10)} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [\tan (x^{e})]$

Step 2.6

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{1}{2}} \ln (10)} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [\tan (x^{e})]$

Step 2.7

Simplify the numerator.

Tap for more steps...

Step 2.7.1

Multiply $- 1$ by $2$ .

$\frac{1}{x^{\frac{1}{2}} \ln (10)} (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [\tan (x^{e})]$

Step 2.7.2

Subtract $2$ from $1$ .

$\frac{1}{x^{\frac{1}{2}} \ln (10)} (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [\tan (x^{e})]$

Step 2.8

Move the negative in front of the fraction.

$\frac{1}{x^{\frac{1}{2}} \ln (10)} (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [\tan (x^{e})]$

Step 2.9

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

$\frac{1}{x^{\frac{1}{2}} \ln (10)} \cdot \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.10

Multiply $\frac{1}{x^{\frac{1}{2}} \ln (10)}$ by $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{x^{- \frac{1}{2}}}{x^{\frac{1}{2}} \ln (10) \cdot 2} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.11

Move $2$ to the left of $x^{\frac{1}{2}} \ln (10)$ .

$\frac{x^{- \frac{1}{2}}}{2 \cdot (x^{\frac{1}{2}} \ln (10))} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.12

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

$\frac{1}{2 x^{\frac{1}{2}} \ln (10) x^{\frac{1}{2}}} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.13

Simplify the denominator.

Tap for more steps...

Step 2.13.1

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

Tap for more steps...

Step 2.13.1.1

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

$\frac{1}{2 (x^{\frac{1}{2}} x^{\frac{1}{2}}) \ln (10)} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.13.1.2

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

$\frac{1}{2 x^{\frac{1}{2} + \frac{1}{2}} \ln (10)} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.13.1.3

Combine the numerators over the common denominator.

$\frac{1}{2 x^{\frac{1 + 1}{2}} \ln (10)} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.13.1.4

Add $1$ and $1$ .

$\frac{1}{2 x^{\frac{2}{2}} \ln (10)} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.13.1.5

Divide $2$ by $2$ .

$\frac{1}{2 x^{1} \ln (10)} + \frac{d}{d x} [\tan (x^{e})]$

Step 2.13.2

Simplify $2 x^{1} \ln (10)$ .

$\frac{1}{2 x \ln (10)} + \frac{d}{d x} [\tan (x^{e})]$

Step 3

Evaluate $\frac{d}{d x} [\tan (x^{e})]$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u_{2}$ as $x^{e}$ .

$\frac{1}{2 x \ln (10)} + \frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [x^{e}]$

Step 3.1.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$\frac{1}{2 x \ln (10)} + \sec^{2} (u_{2}) \frac{d}{d x} [x^{e}]$

Step 3.1.3

Replace all occurrences of $u_{2}$ with $x^{e}$ .

$\frac{1}{2 x \ln (10)} + \sec^{2} (x^{e}) \frac{d}{d x} [x^{e}]$

Step 3.2

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

$\frac{1}{2 x \ln (10)} + \sec^{2} (x^{e}) (e x^{e - 1})$

Step 4

Reorder terms.

$e x^{e - 1} \sec^{2} (x^{e}) + \frac{1}{2 x \ln (10)}$