Calculus Examples

$f (x) = 3 x^{4} - 2 x + \frac{4}{x^{3}} + 8 \sqrt{x} + 6$

Step 1

By the Sum Rule, the derivative of $3 x^{4} - 2 x + \frac{4}{x^{3}} + 8 \sqrt{x} + 6$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 2

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$3 (4 x^{3}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 2.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 3

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

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Step 3.1

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

$12 x^{3} - 2 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 3.2

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

$12 x^{3} - 2 \cdot 1 + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 3.3

Multiply $- 2$ by $1$ .

$12 x^{3} - 2 + \frac{d}{d x} [\frac{4}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4

Evaluate $\frac{d}{d x} [\frac{4}{x^{3}}]$ .

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Step 4.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x^{3}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$12 x^{3} - 2 + 4 \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$12 x^{3} - 2 + 4 \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 4.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$12 x^{3} - 2 + 4 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.3.2

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

$12 x^{3} - 2 + 4 (- u^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$12 x^{3} - 2 + 4 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$12 x^{3} - 2 + 4 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 4.5.1

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

$12 x^{3} - 2 + 4 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.5.2

Multiply $3$ by $- 2$ .

$12 x^{3} - 2 + 4 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.6

Multiply $3$ by $- 1$ .

$12 x^{3} - 2 + 4 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 4.7.1

Move $x^{2}$ .

$12 x^{3} - 2 + 4 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.7.2

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

$12 x^{3} - 2 + 4 (- 3 x^{2 - 6}) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.7.3

Subtract $6$ from $2$ .

$12 x^{3} - 2 + 4 (- 3 x^{- 4}) + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 4.8

Multiply $- 3$ by $4$ .

$12 x^{3} - 2 - 12 x^{- 4} + \frac{d}{d x} [8 \sqrt{x}] + \frac{d}{d x} [6]$

Step 5

Evaluate $\frac{d}{d x} [8 \sqrt{x}]$ .

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Step 5.1

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

$12 x^{3} - 2 - 12 x^{- 4} + \frac{d}{d x} [8 x^{\frac{1}{2}}] + \frac{d}{d x} [6]$

Step 5.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{\frac{1}{2}}$ with respect to $x$ is $8 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$12 x^{3} - 2 - 12 x^{- 4} + 8 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [6]$

Step 5.3

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

$12 x^{3} - 2 - 12 x^{- 4} + 8 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [6]$

Step 5.4

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

$12 x^{3} - 2 - 12 x^{- 4} + 8 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [6]$

Step 5.5

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

$12 x^{3} - 2 - 12 x^{- 4} + 8 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [6]$

Step 5.6

Combine the numerators over the common denominator.

$12 x^{3} - 2 - 12 x^{- 4} + 8 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [6]$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Multiply $- 1$ by $2$ .

$12 x^{3} - 2 - 12 x^{- 4} + 8 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [6]$

Step 5.7.2

Subtract $2$ from $1$ .

$12 x^{3} - 2 - 12 x^{- 4} + 8 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [6]$

Step 5.8

Move the negative in front of the fraction.

$12 x^{3} - 2 - 12 x^{- 4} + 8 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [6]$

Step 5.9

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

$12 x^{3} - 2 - 12 x^{- 4} + 8 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [6]$

Step 5.10

Combine $8$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$12 x^{3} - 2 - 12 x^{- 4} + \frac{8 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [6]$

Step 5.11

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

$12 x^{3} - 2 - 12 x^{- 4} + \frac{8}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [6]$

Step 5.12

Factor $2$ out of $8$ .

$12 x^{3} - 2 - 12 x^{- 4} + \frac{2 \cdot 4}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [6]$

Step 5.13

Cancel the common factors.

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Step 5.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$12 x^{3} - 2 - 12 x^{- 4} + \frac{2 \cdot 4}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [6]$

Step 5.13.2

Cancel the common factor.

$12 x^{3} - 2 - 12 x^{- 4} + \frac{2 \cdot 4}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [6]$

Step 5.13.3

Rewrite the expression.

$12 x^{3} - 2 - 12 x^{- 4} + \frac{4}{x^{\frac{1}{2}}} + \frac{d}{d x} [6]$

Step 6

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$12 x^{3} - 2 - 12 x^{- 4} + \frac{4}{x^{\frac{1}{2}}} + 0$

Step 7

Simplify.

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Step 7.1

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

$12 x^{3} - 2 - 12 \frac{1}{x^{4}} + \frac{4}{x^{\frac{1}{2}}} + 0$

Step 7.2

Combine terms.

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Step 7.2.1

Combine $- 12$ and $\frac{1}{x^{4}}$ .

$12 x^{3} - 2 + \frac{- 12}{x^{4}} + \frac{4}{x^{\frac{1}{2}}} + 0$

Step 7.2.2

Move the negative in front of the fraction.

$12 x^{3} - 2 - \frac{12}{x^{4}} + \frac{4}{x^{\frac{1}{2}}} + 0$

Step 7.2.3

Add $12 x^{3} - 2 - \frac{12}{x^{4}} + \frac{4}{x^{\frac{1}{2}}}$ and $0$ .

$12 x^{3} - 2 - \frac{12}{x^{4}} + \frac{4}{x^{\frac{1}{2}}}$

Step 7.3

Reorder terms.

$12 x^{3} - \frac{12}{x^{4}} - 2 + \frac{4}{x^{\frac{1}{2}}}$