Calculus Examples

$f (x) = 15 \sqrt{x (x - 3)}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x (x - 3)}$ as ${(x (x - 3))}^{\frac{1}{2}}$ .

$\frac{d}{d x} [15 {(x (x - 3))}^{\frac{1}{2}}]$

Step 1.2

Since $15$ is constant with respect to $x$ , the derivative of $15 {(x (x - 3))}^{\frac{1}{2}}$ with respect to $x$ is $15 \frac{d}{d x} [{(x (x - 3))}^{\frac{1}{2}}]$ .

$15 \frac{d}{d x} [{(x (x - 3))}^{\frac{1}{2}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x (x - 3)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x (x - 3)$ .

$15 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x (x - 3)])$

Step 2.2

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

$15 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x (x - 3)])$

Step 2.3

Replace all occurrences of $u$ with $x (x - 3)$ .

$15 (\frac{1}{2} {(x (x - 3))}^{\frac{1}{2} - 1} \frac{d}{d x} [x (x - 3)])$

Step 3

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

$15 (\frac{1}{2} {(x (x - 3))}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x (x - 3)])$

Step 4

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

$15 (\frac{1}{2} {(x (x - 3))}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x (x - 3)])$

Step 5

Combine the numerators over the common denominator.

$15 (\frac{1}{2} {(x (x - 3))}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x (x - 3)])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$15 (\frac{1}{2} {(x (x - 3))}^{\frac{1 - 2}{2}} \frac{d}{d x} [x (x - 3)])$

Step 6.2

Subtract $2$ from $1$ .

$15 (\frac{1}{2} {(x (x - 3))}^{\frac{- 1}{2}} \frac{d}{d x} [x (x - 3)])$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$15 (\frac{1}{2} {(x (x - 3))}^{- \frac{1}{2}} \frac{d}{d x} [x (x - 3)])$

Step 7.2

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

$15 (\frac{{(x (x - 3))}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x (x - 3)])$

Step 7.3

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

$15 (\frac{1}{2 {(x (x - 3))}^{\frac{1}{2}}} \frac{d}{d x} [x (x - 3)])$

Step 7.4

Combine $\frac{1}{2 {(x (x - 3))}^{\frac{1}{2}}}$ and $15$ .

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} \frac{d}{d x} [x (x - 3)]$

Step 8

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 $f (x) = x$ and $g (x) = x - 3$ .

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x \frac{d}{d x} [x - 3] + (x - 3) \frac{d}{d x} [x])$

Step 9

Differentiate.

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Step 9.1

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

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [x])$

Step 9.2

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

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x (1 + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [x])$

Step 9.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x (1 + 0) + (x - 3) \frac{d}{d x} [x])$

Step 9.4

Simplify the expression.

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Step 9.4.1

Add $1$ and $0$ .

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x \cdot 1 + (x - 3) \frac{d}{d x} [x])$

Step 9.4.2

Multiply $x$ by $1$ .

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x + (x - 3) \frac{d}{d x} [x])$

Step 9.5

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

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x + (x - 3) \cdot 1)$

Step 9.6

Simplify by adding terms.

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Step 9.6.1

Multiply $x - 3$ by $1$ .

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (x + x - 3)$

Step 9.6.2

Add $x$ and $x$ .

$\frac{15}{2 {(x (x - 3))}^{\frac{1}{2}}} (2 x - 3)$

Step 10

Simplify.

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Step 10.1

Apply the product rule to $x (x - 3)$ .

$\frac{15}{2 (x^{\frac{1}{2}} {(x - 3)}^{\frac{1}{2}})} (2 x - 3)$

Step 10.2

Reorder the factors of $\frac{15}{2 x^{\frac{1}{2}} {(x - 3)}^{\frac{1}{2}}} (2 x - 3)$ .

$(2 x - 3) \frac{15}{2 x^{\frac{1}{2}} {(x - 3)}^{\frac{1}{2}}}$