Calculus Examples

$y = (9 \sqrt{x} + 8) x^{2}$

Step 1

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

$\frac{d}{d x} [(9 x^{\frac{1}{2}} + 8) x^{2}]$

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 $f (x) = 9 x^{\frac{1}{2}} + 8$ and $g (x) = x^{2}$ .

$(9 x^{\frac{1}{2}} + 8) \frac{d}{d x} [x^{2}] + x^{2} \frac{d}{d x} [9 x^{\frac{1}{2}} + 8]$

Step 3

Differentiate.

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Step 3.1

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

$(9 x^{\frac{1}{2}} + 8) (2 x) + x^{2} \frac{d}{d x} [9 x^{\frac{1}{2}} + 8]$

Step 3.2

Move $2$ to the left of $9 x^{\frac{1}{2}} + 8$ .

$2 \cdot (9 x^{\frac{1}{2}} + 8) x + x^{2} \frac{d}{d x} [9 x^{\frac{1}{2}} + 8]$

Step 3.3

By the Sum Rule, the derivative of $9 x^{\frac{1}{2}} + 8$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{1}{2}}] + \frac{d}{d x} [8]$ .

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (\frac{d}{d x} [9 x^{\frac{1}{2}}] + \frac{d}{d x} [8])$

Step 3.4

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [8])$

Step 3.5

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [8])$

Step 4

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [8])$

Step 5

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [8])$

Step 6

Combine the numerators over the common denominator.

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [8])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [8])$

Step 7.2

Subtract $2$ from $1$ .

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [8])$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [8])$

Step 8.2

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (9 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [8])$

Step 8.3

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (\frac{9 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [8])$

Step 8.4

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (\frac{9}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [8])$

Step 9

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} (\frac{9}{2 x^{\frac{1}{2}}} + 0)$

Step 10

Combine fractions.

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Step 10.1

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

$2 (9 x^{\frac{1}{2}} + 8) x + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

Step 10.2

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

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{x^{2} \cdot 9}{2 x^{\frac{1}{2}}}$

Step 10.3

Simplify the expression.

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Step 10.3.1

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

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 \cdot x^{2}}{2 x^{\frac{1}{2}}}$

Step 10.3.2

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

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 x^{2} x^{- \frac{1}{2}}}{2}$

Step 11

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

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Step 11.1

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

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 (x^{- \frac{1}{2}} x^{2})}{2}$

Step 11.2

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

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 x^{- \frac{1}{2} + 2}}{2}$

Step 11.3

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

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 x^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}{2}$

Step 11.4

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

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 x^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}{2}$

Step 11.5

Combine the numerators over the common denominator.

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 x^{\frac{- 1 + 2 \cdot 2}{2}}}{2}$

Step 11.6

Simplify the numerator.

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Step 11.6.1

Multiply $2$ by $2$ .

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 x^{\frac{- 1 + 4}{2}}}{2}$

Step 11.6.2

Add $- 1$ and $4$ .

$2 (9 x^{\frac{1}{2}} + 8) x + \frac{9 x^{\frac{3}{2}}}{2}$

Step 12

Combine $2 (9 x^{\frac{1}{2}} + 8) x$ and $\frac{9 x^{\frac{3}{2}}}{2}$ using a common denominator.

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Step 12.1

Move $9 x^{\frac{1}{2}} + 8$ .

$2 x (9 x^{\frac{1}{2}} + 8) + \frac{9 x^{\frac{3}{2}}}{2}$

Step 12.2

To write $2 x (9 x^{\frac{1}{2}} + 8)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x (9 x^{\frac{1}{2}} + 8) \cdot \frac{2}{2} + \frac{9 x^{\frac{3}{2}}}{2}$

Step 12.3

Combine $2 x (9 x^{\frac{1}{2}} + 8)$ and $\frac{2}{2}$ .

$\frac{2 x (9 x^{\frac{1}{2}} + 8) \cdot 2}{2} + \frac{9 x^{\frac{3}{2}}}{2}$

Step 12.4

Combine the numerators over the common denominator.

$\frac{2 x (9 x^{\frac{1}{2}} + 8) \cdot 2 + 9 x^{\frac{3}{2}}}{2}$

Step 13

Multiply $2$ by $2$ .

$\frac{4 x (9 x^{\frac{1}{2}} + 8) + 9 x^{\frac{3}{2}}}{2}$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{4 x (9 x^{\frac{1}{2}}) + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2

Simplify the numerator.

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Step 14.2.1

Simplify each term.

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Step 14.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 \cdot 9 x \cdot x^{\frac{1}{2}} + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.2

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

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Step 14.2.1.2.1

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

$\frac{4 \cdot 9 (x^{\frac{1}{2}} x) + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.2.2

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

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Step 14.2.1.2.2.1

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

$\frac{4 \cdot 9 (x^{\frac{1}{2}} x^{1}) + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.2.2.2

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

$\frac{4 \cdot 9 x^{\frac{1}{2} + 1} + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.2.3

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

$\frac{4 \cdot 9 x^{\frac{1}{2} + \frac{2}{2}} + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.2.4

Combine the numerators over the common denominator.

$\frac{4 \cdot 9 x^{\frac{1 + 2}{2}} + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.2.5

Add $1$ and $2$ .

$\frac{4 \cdot 9 x^{\frac{3}{2}} + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.3

Multiply $4$ by $9$ .

$\frac{36 x^{\frac{3}{2}} + 4 x \cdot 8 + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.1.4

Multiply $8$ by $4$ .

$\frac{36 x^{\frac{3}{2}} + 32 x + 9 x^{\frac{3}{2}}}{2}$

Step 14.2.2

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

$\frac{32 x + 45 x^{\frac{3}{2}}}{2}$

Step 14.3

Factor $x$ out of $32 x + 45 x^{\frac{3}{2}}$ .

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Step 14.3.1

Factor $x$ out of $32 x$ .

$\frac{x \cdot 32 + 45 x^{\frac{3}{2}}}{2}$

Step 14.3.2

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

$\frac{x \cdot 32 + x (45 x^{\frac{1}{2}})}{2}$

Step 14.3.3

Factor $x$ out of $x \cdot 32 + x (45 x^{\frac{1}{2}})$ .

$\frac{x (32 + 45 x^{\frac{1}{2}})}{2}$