Calculus Examples

$f (x) = (x^{2}) (9 \sqrt{x} + 5)$

Step 1

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

$\frac{d}{d x} [x^{2} (9 x^{\frac{1}{2}} + 5)]$

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 $g (x) = 9 x^{\frac{1}{2}} + 5$ .

$x^{2} \frac{d}{d x} [9 x^{\frac{1}{2}} + 5] + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $9 x^{\frac{1}{2}} + 5$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{1}{2}}] + \frac{d}{d x} [5]$ .

$x^{2} (\frac{d}{d x} [9 x^{\frac{1}{2}}] + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 3.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{1}{2}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$x^{2} (9 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 3.3

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

$x^{2} (9 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 4

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

$x^{2} (9 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 5

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

$x^{2} (9 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 6

Combine the numerators over the common denominator.

$x^{2} (9 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$x^{2} (9 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 7.2

Subtract $2$ from $1$ .

$x^{2} (9 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$x^{2} (9 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 8.2

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

$x^{2} (9 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 8.3

Combine $9$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$x^{2} (\frac{9 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 8.4

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

$x^{2} (\frac{9}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [5]) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 9

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$x^{2} (\frac{9}{2 x^{\frac{1}{2}}} + 0) + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 10

Combine fractions.

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Step 10.1

Add $\frac{9}{2 x^{\frac{1}{2}}}$ and $0$ .

$x^{2} \frac{9}{2 x^{\frac{1}{2}}} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 10.2

Combine $x^{2}$ and $\frac{9}{2 x^{\frac{1}{2}}}$ .

$\frac{x^{2} \cdot 9}{2 x^{\frac{1}{2}}} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 10.3

Simplify the expression.

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Step 10.3.1

Move $9$ to the left of $x^{2}$ .

$\frac{9 \cdot x^{2}}{2 x^{\frac{1}{2}}} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 10.3.2

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

$\frac{9 x^{2} x^{- \frac{1}{2}}}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 11

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

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Step 11.1

Move $x^{- \frac{1}{2}}$ .

$\frac{9 (x^{- \frac{1}{2}} x^{2})}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 11.2

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

$\frac{9 x^{- \frac{1}{2} + 2}}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 11.3

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

$\frac{9 x^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 11.4

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

$\frac{9 x^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 11.5

Combine the numerators over the common denominator.

$\frac{9 x^{\frac{- 1 + 2 \cdot 2}{2}}}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 11.6

Simplify the numerator.

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Step 11.6.1

Multiply $2$ by $2$ .

$\frac{9 x^{\frac{- 1 + 4}{2}}}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 11.6.2

Add $- 1$ and $4$ .

$\frac{9 x^{\frac{3}{2}}}{2} + (9 x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{2}]$

Step 12

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

$\frac{9 x^{\frac{3}{2}}}{2} + (9 x^{\frac{1}{2}} + 5) (2 x)$

Step 13

Move $2$ to the left of $9 x^{\frac{1}{2}} + 5$ .

$\frac{9 x^{\frac{3}{2}}}{2} + 2 \cdot (9 x^{\frac{1}{2}} + 5) x$

Step 14

Combine $\frac{9 x^{\frac{3}{2}}}{2}$ and $2 (9 x^{\frac{1}{2}} + 5) x$ using a common denominator.

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Step 14.1

Move $9 x^{\frac{1}{2}} + 5$ .

$\frac{9 x^{\frac{3}{2}}}{2} + 2 x (9 x^{\frac{1}{2}} + 5)$

Step 14.2

To write $2 x (9 x^{\frac{1}{2}} + 5)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9 x^{\frac{3}{2}}}{2} + 2 x (9 x^{\frac{1}{2}} + 5) \cdot \frac{2}{2}$

Step 14.3

Combine $2 x (9 x^{\frac{1}{2}} + 5)$ and $\frac{2}{2}$ .

$\frac{9 x^{\frac{3}{2}}}{2} + \frac{2 x (9 x^{\frac{1}{2}} + 5) \cdot 2}{2}$

Step 14.4

Combine the numerators over the common denominator.

$\frac{9 x^{\frac{3}{2}} + 2 x (9 x^{\frac{1}{2}} + 5) \cdot 2}{2}$

Step 15

Multiply $2$ by $2$ .

$\frac{9 x^{\frac{3}{2}} + 4 x (9 x^{\frac{1}{2}} + 5)}{2}$

Step 16

Simplify.

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Step 16.1

Apply the distributive property.

$\frac{9 x^{\frac{3}{2}} + 4 x (9 x^{\frac{1}{2}}) + 4 x \cdot 5}{2}$

Step 16.2

Simplify the numerator.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{9 x^{\frac{3}{2}} + 4 \cdot 9 x \cdot x^{\frac{1}{2}} + 4 x \cdot 5}{2}$

Step 16.2.1.2

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

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Step 16.2.1.2.1

Move $x^{\frac{1}{2}}$ .

$\frac{9 x^{\frac{3}{2}} + 4 \cdot 9 (x^{\frac{1}{2}} x) + 4 x \cdot 5}{2}$

Step 16.2.1.2.2

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

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Step 16.2.1.2.2.1

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

$\frac{9 x^{\frac{3}{2}} + 4 \cdot 9 (x^{\frac{1}{2}} x^{1}) + 4 x \cdot 5}{2}$

Step 16.2.1.2.2.2

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

$\frac{9 x^{\frac{3}{2}} + 4 \cdot 9 x^{\frac{1}{2} + 1} + 4 x \cdot 5}{2}$

Step 16.2.1.2.3

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

$\frac{9 x^{\frac{3}{2}} + 4 \cdot 9 x^{\frac{1}{2} + \frac{2}{2}} + 4 x \cdot 5}{2}$

Step 16.2.1.2.4

Combine the numerators over the common denominator.

$\frac{9 x^{\frac{3}{2}} + 4 \cdot 9 x^{\frac{1 + 2}{2}} + 4 x \cdot 5}{2}$

Step 16.2.1.2.5

Add $1$ and $2$ .

$\frac{9 x^{\frac{3}{2}} + 4 \cdot 9 x^{\frac{3}{2}} + 4 x \cdot 5}{2}$

Step 16.2.1.3

Multiply $4$ by $9$ .

$\frac{9 x^{\frac{3}{2}} + 36 x^{\frac{3}{2}} + 4 x \cdot 5}{2}$

Step 16.2.1.4

Multiply $5$ by $4$ .

$\frac{9 x^{\frac{3}{2}} + 36 x^{\frac{3}{2}} + 20 x}{2}$

Step 16.2.2

Add $9 x^{\frac{3}{2}}$ and $36 x^{\frac{3}{2}}$ .

$\frac{45 x^{\frac{3}{2}} + 20 x}{2}$

Step 16.3

Reorder terms.

$\frac{20 x + 45 x^{\frac{3}{2}}}{2}$

Step 16.4

Factor $5 x$ out of $20 x + 45 x^{\frac{3}{2}}$ .

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Step 16.4.1

Factor $5 x$ out of $20 x$ .

$\frac{5 x (4) + 45 x^{\frac{3}{2}}}{2}$

Step 16.4.2

Factor $5 x$ out of $45 x^{\frac{3}{2}}$ .

$\frac{5 x (4) + 5 x (9 x^{\frac{1}{2}})}{2}$

Step 16.4.3

Factor $5 x$ out of $5 x (4) + 5 x (9 x^{\frac{1}{2}})$ .

$\frac{5 x (4 + 9 x^{\frac{1}{2}})}{2}$