Calculus Examples

$p (x) = x^{2} \sqrt{x^{4} - 12}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} - 12}$ as ${(x^{4} - 12)}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = {(x^{4} - 12)}^{\frac{1}{2}}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{4} - 12$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{4} - 12$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x^{4} - 12$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

Combine $\frac{1}{2}$ and ${(x^{4} - 12)}^{- \frac{1}{2}}$ .

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

Step 8.3

Move ${(x^{4} - 12)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.4

Combine $\frac{1}{2 {(x^{4} - 12)}^{\frac{1}{2}}}$ and $x^{2}$ .

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

Step 9

By the Sum Rule, the derivative of $x^{4} - 12$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12]$ .

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

Step 10

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

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

Step 11

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$\frac{x^{2}}{2 {(x^{4} - 12)}^{\frac{1}{2}}} (4 x^{3} + 0) + {(x^{4} - 12)}^{\frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 12

Combine fractions.

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Step 12.1

Add $4 x^{3}$ and $0$ .

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

Step 12.2

Combine $4$ and $\frac{x^{2}}{2 {(x^{4} - 12)}^{\frac{1}{2}}}$ .

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

Step 12.3

Combine $\frac{4 x^{2}}{2 {(x^{4} - 12)}^{\frac{1}{2}}}$ and $x^{3}$ .

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

Step 13

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 13.1

Move $x^{3}$ .

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

Step 13.2

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

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

Step 13.3

Add $3$ and $2$ .

$\frac{4 x^{5}}{2 {(x^{4} - 12)}^{\frac{1}{2}}} + {(x^{4} - 12)}^{\frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 14

Factor $2$ out of $4 x^{5}$ .

$\frac{2 (2 x^{5})}{2 {(x^{4} - 12)}^{\frac{1}{2}}} + {(x^{4} - 12)}^{\frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 15

Cancel the common factors.

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Step 15.1

Factor $2$ out of $2 {(x^{4} - 12)}^{\frac{1}{2}}$ .

$\frac{2 (2 x^{5})}{2 ({(x^{4} - 12)}^{\frac{1}{2}})} + {(x^{4} - 12)}^{\frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 15.2

Cancel the common factor.

$\frac{2 (2 x^{5})}{2 {(x^{4} - 12)}^{\frac{1}{2}}} + {(x^{4} - 12)}^{\frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 15.3

Rewrite the expression.

$\frac{2 x^{5}}{{(x^{4} - 12)}^{\frac{1}{2}}} + {(x^{4} - 12)}^{\frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 16

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

$\frac{2 x^{5}}{{(x^{4} - 12)}^{\frac{1}{2}}} + {(x^{4} - 12)}^{\frac{1}{2}} (2 x)$

Step 17

Move $2$ to the left of ${(x^{4} - 12)}^{\frac{1}{2}}$ .

$\frac{2 x^{5}}{{(x^{4} - 12)}^{\frac{1}{2}}} + 2 \cdot {(x^{4} - 12)}^{\frac{1}{2}} x$

Step 18

Combine $\frac{2 x^{5}}{{(x^{4} - 12)}^{\frac{1}{2}}}$ and $2 {(x^{4} - 12)}^{\frac{1}{2}} x$ using a common denominator.

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Step 18.1

Move ${(x^{4} - 12)}^{\frac{1}{2}}$ .

$\frac{2 x^{5}}{{(x^{4} - 12)}^{\frac{1}{2}}} + 2 x {(x^{4} - 12)}^{\frac{1}{2}}$

Step 18.2

To write $2 x {(x^{4} - 12)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{4} - 12)}^{\frac{1}{2}}}{{(x^{4} - 12)}^{\frac{1}{2}}}$ .

$\frac{2 x^{5}}{{(x^{4} - 12)}^{\frac{1}{2}}} + \frac{2 x {(x^{4} - 12)}^{\frac{1}{2}} {(x^{4} - 12)}^{\frac{1}{2}}}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 18.3

Combine the numerators over the common denominator.

$\frac{2 x^{5} + 2 x {(x^{4} - 12)}^{\frac{1}{2}} {(x^{4} - 12)}^{\frac{1}{2}}}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 19

Multiply ${(x^{4} - 12)}^{\frac{1}{2}}$ by ${(x^{4} - 12)}^{\frac{1}{2}}$ by adding the exponents.

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Step 19.1

Move ${(x^{4} - 12)}^{\frac{1}{2}}$ .

$\frac{2 x^{5} + 2 x ({(x^{4} - 12)}^{\frac{1}{2}} {(x^{4} - 12)}^{\frac{1}{2}})}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 19.2

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

$\frac{2 x^{5} + 2 x {(x^{4} - 12)}^{\frac{1}{2} + \frac{1}{2}}}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 19.3

Combine the numerators over the common denominator.

$\frac{2 x^{5} + 2 x {(x^{4} - 12)}^{\frac{1 + 1}{2}}}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 19.4

Add $1$ and $1$ .

$\frac{2 x^{5} + 2 x {(x^{4} - 12)}^{\frac{2}{2}}}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 19.5

Divide $2$ by $2$ .

$\frac{2 x^{5} + 2 x {(x^{4} - 12)}^{1}}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 20

Simplify $2 x {(x^{4} - 12)}^{1}$ .

$\frac{2 x^{5} + 2 x (x^{4} - 12)}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21

Simplify.

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Step 21.1

Apply the distributive property.

$\frac{2 x^{5} + 2 x \cdot x^{4} + 2 x \cdot - 12}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.2

Simplify the numerator.

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Step 21.2.1

Simplify each term.

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Step 21.2.1.1

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 21.2.1.1.1

Move $x^{4}$ .

$\frac{2 x^{5} + 2 (x^{4} x) + 2 x \cdot - 12}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.2.1.1.2

Multiply $x^{4}$ by $x$ .

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Step 21.2.1.1.2.1

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

$\frac{2 x^{5} + 2 (x^{4} x^{1}) + 2 x \cdot - 12}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.2.1.1.2.2

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

$\frac{2 x^{5} + 2 x^{4 + 1} + 2 x \cdot - 12}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.2.1.1.3

Add $4$ and $1$ .

$\frac{2 x^{5} + 2 x^{5} + 2 x \cdot - 12}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.2.1.2

Multiply $- 12$ by $2$ .

$\frac{2 x^{5} + 2 x^{5} - 24 x}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.2.2

Add $2 x^{5}$ and $2 x^{5}$ .

$\frac{4 x^{5} - 24 x}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.3

Factor $4 x$ out of $4 x^{5} - 24 x$ .

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Step 21.3.1

Factor $4 x$ out of $4 x^{5}$ .

$\frac{4 x (x^{4}) - 24 x}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.3.2

Factor $4 x$ out of $- 24 x$ .

$\frac{4 x (x^{4}) + 4 x (- 6)}{{(x^{4} - 12)}^{\frac{1}{2}}}$

Step 21.3.3

Factor $4 x$ out of $4 x (x^{4}) + 4 x (- 6)$ .

$\frac{4 x (x^{4} - 6)}{{(x^{4} - 12)}^{\frac{1}{2}}}$