Calculus Examples

$f (x) = 4 x^{2} + 3 \sqrt{x} + \frac{6}{\sqrt{x}}$

Step 1

By the Sum Rule, the derivative of $4 x^{2} + 3 \sqrt{x} + \frac{6}{\sqrt{x}}$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 2.2

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

$4 (2 x) + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3

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

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Step 3.1

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

$8 x + \frac{d}{d x} [3 x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.2

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

$8 x + 3 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.3

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

$8 x + 3 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.4

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

$8 x + 3 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.5

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

$8 x + 3 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.6

Combine the numerators over the common denominator.

$8 x + 3 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$8 x + 3 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.7.2

Subtract $2$ from $1$ .

$8 x + 3 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.8

Move the negative in front of the fraction.

$8 x + 3 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.9

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

$8 x + 3 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.10

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

$8 x + \frac{3 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 3.11

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [\frac{6}{\sqrt{x}}]$

Step 4

Evaluate $\frac{d}{d x} [\frac{6}{\sqrt{x}}]$ .

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Step 4.1

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [\frac{6}{x^{\frac{1}{2}}}]$

Step 4.2

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x^{\frac{1}{2}}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 4.3

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]$

Step 4.4

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

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Step 4.4.1

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.4.2

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.4.3

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.5

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 4.6

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

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Step 4.6.1

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 4.6.2

Cancel the common factor of $2$ .

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Step 4.6.2.1

Factor $2$ out of $- 2$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 4.6.2.2

Cancel the common factor.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 4.6.2.3

Rewrite the expression.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 4.7

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 4.8

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 4.9

Combine the numerators over the common denominator.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 4.10

Simplify the numerator.

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Step 4.10.1

Multiply $- 1$ by $2$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 4.10.2

Subtract $2$ from $1$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 4.11

Move the negative in front of the fraction.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 4.12

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 4.13

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 4.14

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

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Step 4.14.1

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 4.14.2

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 4.14.3

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 4.14.4

Combine the numerators over the common denominator.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 4.14.5

Simplify the numerator.

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Step 4.14.5.1

Multiply $- 1$ by $2$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 4.14.5.2

Subtract $2$ from $- 1$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{\frac{- 3}{2}}}{2})$

Step 4.14.6

Move the negative in front of the fraction.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 4.15

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

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + 6 (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 4.16

Multiply $- 1$ by $6$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} - 6 \frac{1}{2 x^{\frac{3}{2}}}$

Step 4.17

Combine $- 6$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + \frac{- 6}{2 x^{\frac{3}{2}}}$

Step 4.18

Factor $2$ out of $- 6$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + \frac{2 \cdot - 3}{2 x^{\frac{3}{2}}}$

Step 4.19

Cancel the common factors.

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Step 4.19.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + \frac{2 \cdot - 3}{2 (x^{\frac{3}{2}})}$

Step 4.19.2

Cancel the common factor.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + \frac{2 \cdot - 3}{2 x^{\frac{3}{2}}}$

Step 4.19.3

Rewrite the expression.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} + \frac{- 3}{x^{\frac{3}{2}}}$

Step 4.20

Move the negative in front of the fraction.

$8 x + \frac{3}{2 x^{\frac{1}{2}}} - \frac{3}{x^{\frac{3}{2}}}$