Calculus Examples

$y = \sqrt{x} + \frac{3}{\sqrt{2 x}}$

Step 1

By the Sum Rule, the derivative of $\sqrt{x} + \frac{3}{\sqrt{2 x}}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$ .

$\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2

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

Tap for more steps...

Step 2.1

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

$\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2.2

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

$\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2.3

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2.4

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

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 2.7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{3}{\sqrt{2 x}}]$ .

Tap for more steps...

Step 3.1

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

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [\frac{3}{{(2 x)}^{\frac{1}{2}}}]$

Step 3.2

Factor $2$ out of $2 x$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [\frac{3}{{(2 (x))}^{\frac{1}{2}}}]$

Step 3.3

Apply the product rule to $2 (x)$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [\frac{3}{2^{\frac{1}{2}} x^{\frac{1}{2}}}]$

Step 3.4

Since $\frac{3}{2^{\frac{1}{2}}}$ is constant with respect to $x$ , the derivative of $\frac{3}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{3}{2^{\frac{1}{2}}} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{3}{2^{\frac{1}{2}}} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 3.5

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

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{3}{2^{\frac{1}{2}}} \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{3}{2^{\frac{1}{2}}} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.6.2

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

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{3}{2^{\frac{1}{2}}} (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.6.3

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

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{3}{2^{\frac{1}{2}}} (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.7

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

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

Step 3.8

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

Tap for more steps...

Step 3.8.1

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

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

Step 3.8.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.8.2.1

Factor $2$ out of $- 2$ .

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

Step 3.8.2.2

Cancel the common factor.

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

Step 3.8.2.3

Rewrite the expression.

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Combine the numerators over the common denominator.

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

Step 3.12

Simplify the numerator.

Tap for more steps...

Step 3.12.1

Multiply $- 1$ by $2$ .

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

Step 3.12.2

Subtract $2$ from $1$ .

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

Step 3.13

Move the negative in front of the fraction.

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

Step 3.14

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

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

Step 3.15

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

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

Step 3.16

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

Tap for more steps...

Step 3.16.1

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

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

Step 3.16.2

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

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

Step 3.16.3

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

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

Step 3.16.4

Combine the numerators over the common denominator.

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

Step 3.16.5

Simplify the numerator.

Tap for more steps...

Step 3.16.5.1

Multiply $- 1$ by $2$ .

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

Step 3.16.5.2

Subtract $2$ from $- 1$ .

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

Step 3.16.6

Move the negative in front of the fraction.

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

Step 3.17

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

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

Step 3.18

Multiply $\frac{3}{2^{\frac{1}{2}}}$ by $\frac{1}{2 x^{\frac{3}{2}}}$ .

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

Step 3.19

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

Tap for more steps...

Step 3.19.1

Move $2$ .

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{3}{2 \cdot 2^{\frac{1}{2}} x^{\frac{3}{2}}}$

Step 3.19.2

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

Tap for more steps...

Step 3.19.2.1

Raise $2$ to the power of $1$ .

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{3}{2^{1} \cdot 2^{\frac{1}{2}} x^{\frac{3}{2}}}$

Step 3.19.2.2

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

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{3}{2^{1 + \frac{1}{2}} x^{\frac{3}{2}}}$

Step 3.19.3

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

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{3}{2^{\frac{2}{2} + \frac{1}{2}} x^{\frac{3}{2}}}$

Step 3.19.4

Combine the numerators over the common denominator.

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{3}{2^{\frac{2 + 1}{2}} x^{\frac{3}{2}}}$

Step 3.19.5

Add $2$ and $1$ .

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{3}{2^{\frac{3}{2}} x^{\frac{3}{2}}}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} - \frac{3}{2^{\frac{3}{2}} x^{\frac{3}{2}}}$

Step 4.2

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

$\frac{1}{2 x^{\frac{1}{2}}} - \frac{3}{2^{\frac{3}{2}} x^{\frac{3}{2}}}$