Find the Derivative Using Product Rule - d/dx y=(9 square root of x+2)x^2

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

Step 1

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

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

Step 2

Differentiate.

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

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

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

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

Step 3

Evaluate $\frac{d}{d x} [9 \sqrt{x}]$ .

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Step 4

Differentiate using the Constant Rule.

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

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

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

Step 5

Simplify.

Apply the distributive property.

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

Combine terms.

Multiply $2$ by $9$ .

$18 \sqrt{x} x + 2 (2 x) + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

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

$18 x^{\frac{1}{2}} x + 2 (2 x) + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

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

$18 (x^{1} x^{\frac{1}{2}}) + 2 (2 x) + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

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

$18 x^{1 + \frac{1}{2}} + 2 (2 x) + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

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

$18 x^{\frac{2}{2} + \frac{1}{2}} + 2 (2 x) + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

Combine the numerators over the common denominator.

$18 x^{\frac{2 + 1}{2}} + 2 (2 x) + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

Add $2$ and $1$ .

$18 x^{\frac{3}{2}} + 2 (2 x) + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

Multiply $2$ by $2$ .

$18 x^{\frac{3}{2}} + 4 x + x^{2} \frac{9}{2 x^{\frac{1}{2}}}$

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

$18 x^{\frac{3}{2}} + 4 x + \frac{x^{2} \cdot 9}{2 x^{\frac{1}{2}}}$

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

$18 x^{\frac{3}{2}} + 4 x + \frac{9 \cdot x^{2}}{2 x^{\frac{1}{2}}}$

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

$18 x^{\frac{3}{2}} + 4 x + \frac{9 x^{2} x^{- \frac{1}{2}}}{2}$

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

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

$18 x^{\frac{3}{2}} + 4 x + \frac{9 (x^{- \frac{1}{2}} x^{2})}{2}$

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

$18 x^{\frac{3}{2}} + 4 x + \frac{9 x^{- \frac{1}{2} + 2}}{2}$

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

$18 x^{\frac{3}{2}} + 4 x + \frac{9 x^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}{2}$

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

$18 x^{\frac{3}{2}} + 4 x + \frac{9 x^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}{2}$

Combine the numerators over the common denominator.

$18 x^{\frac{3}{2}} + 4 x + \frac{9 x^{\frac{- 1 + 2 \cdot 2}{2}}}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$18 x^{\frac{3}{2}} + 4 x + \frac{9 x^{\frac{- 1 + 4}{2}}}{2}$

Add $- 1$ and $4$ .

$18 x^{\frac{3}{2}} + 4 x + \frac{9 x^{\frac{3}{2}}}{2}$

To write $18 x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 x + 18 x^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{9 x^{\frac{3}{2}}}{2}$

Combine $18 x^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$4 x + \frac{18 x^{\frac{3}{2}} \cdot 2}{2} + \frac{9 x^{\frac{3}{2}}}{2}$

Combine the numerators over the common denominator.

$4 x + \frac{18 x^{\frac{3}{2}} \cdot 2 + 9 x^{\frac{3}{2}}}{2}$

Multiply $2$ by $18$ .

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

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

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

Find the Derivative Using Product Rule - d/dx y=(9 square root of x+2)x^2

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