Calculus Examples

$y = (7 \sqrt{x} + 4) x^{2}$

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 $f (x) = 7 \sqrt{x} + 4$ and $g (x) = x^{2}$ .

$(7 \sqrt{x} + 4) \frac{d}{d x} [x^{2}] + x^{2} \frac{d}{d x} [7 \sqrt{x} + 4]$

Step 2

Differentiate.

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Step 2.1

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

$(7 \sqrt{x} + 4) (2 x) + x^{2} \frac{d}{d x} [7 \sqrt{x} + 4]$

Step 2.2

By the Sum Rule, the derivative of $7 \sqrt{x} + 4$ with respect to $x$ is $\frac{d}{d x} [7 \sqrt{x}] + \frac{d}{d x} [4]$ .

$(7 \sqrt{x} + 4) (2 x) + x^{2} (\frac{d}{d x} [7 \sqrt{x}] + \frac{d}{d x} [4])$

Step 3

Evaluate $\frac{d}{d x} [7 \sqrt{x}]$ .

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Step 3.1

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

$(7 \sqrt{x} + 4) (2 x) + x^{2} (\frac{d}{d x} [7 x^{\frac{1}{2}}] + \frac{d}{d x} [4])$

Step 3.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{\frac{1}{2}}$ with respect to $x$ is $7 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [4])$

Step 3.3

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

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [4])$

Step 3.4

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

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [4])$

Step 3.5

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

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [4])$

Step 3.6

Combine the numerators over the common denominator.

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [4])$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [4])$

Step 3.7.2

Subtract $2$ from $1$ .

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [4])$

Step 3.8

Move the negative in front of the fraction.

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [4])$

Step 3.9

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

$(7 \sqrt{x} + 4) (2 x) + x^{2} (7 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [4])$

Step 3.10

Combine $7$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$(7 \sqrt{x} + 4) (2 x) + x^{2} (\frac{7 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [4])$

Step 3.11

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

$(7 \sqrt{x} + 4) (2 x) + x^{2} (\frac{7}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [4])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$(7 \sqrt{x} + 4) (2 x) + x^{2} (\frac{7}{2 x^{\frac{1}{2}}} + 0)$

Step 4.2

Add $\frac{7}{2 x^{\frac{1}{2}}}$ and $0$ .

$(7 \sqrt{x} + 4) (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$7 \sqrt{x} (2 x) + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $2$ by $7$ .

$14 \sqrt{x} x + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.2

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

$14 x^{\frac{1}{2}} x + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.3

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

$14 (x^{1} x^{\frac{1}{2}}) + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.4

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

$14 x^{1 + \frac{1}{2}} + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.5

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

$14 x^{\frac{2}{2} + \frac{1}{2}} + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.6

Combine the numerators over the common denominator.

$14 x^{\frac{2 + 1}{2}} + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.7

Add $2$ and $1$ .

$14 x^{\frac{3}{2}} + 4 (2 x) + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.8

Multiply $2$ by $4$ .

$14 x^{\frac{3}{2}} + 8 x + x^{2} \frac{7}{2 x^{\frac{1}{2}}}$

Step 5.2.9

Combine $x^{2}$ and $\frac{7}{2 x^{\frac{1}{2}}}$ .

$14 x^{\frac{3}{2}} + 8 x + \frac{x^{2} \cdot 7}{2 x^{\frac{1}{2}}}$

Step 5.2.10

Move $7$ to the left of $x^{2}$ .

$14 x^{\frac{3}{2}} + 8 x + \frac{7 \cdot x^{2}}{2 x^{\frac{1}{2}}}$

Step 5.2.11

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

$14 x^{\frac{3}{2}} + 8 x + \frac{7 x^{2} x^{- \frac{1}{2}}}{2}$

Step 5.2.12

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

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Step 5.2.12.1

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

$14 x^{\frac{3}{2}} + 8 x + \frac{7 (x^{- \frac{1}{2}} x^{2})}{2}$

Step 5.2.12.2

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

$14 x^{\frac{3}{2}} + 8 x + \frac{7 x^{- \frac{1}{2} + 2}}{2}$

Step 5.2.12.3

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

$14 x^{\frac{3}{2}} + 8 x + \frac{7 x^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}{2}$

Step 5.2.12.4

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

$14 x^{\frac{3}{2}} + 8 x + \frac{7 x^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}{2}$

Step 5.2.12.5

Combine the numerators over the common denominator.

$14 x^{\frac{3}{2}} + 8 x + \frac{7 x^{\frac{- 1 + 2 \cdot 2}{2}}}{2}$

Step 5.2.12.6

Simplify the numerator.

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Step 5.2.12.6.1

Multiply $2$ by $2$ .

$14 x^{\frac{3}{2}} + 8 x + \frac{7 x^{\frac{- 1 + 4}{2}}}{2}$

Step 5.2.12.6.2

Add $- 1$ and $4$ .

$14 x^{\frac{3}{2}} + 8 x + \frac{7 x^{\frac{3}{2}}}{2}$

Step 5.2.13

To write $14 x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 x + 14 x^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{7 x^{\frac{3}{2}}}{2}$

Step 5.2.14

Combine $14 x^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$8 x + \frac{14 x^{\frac{3}{2}} \cdot 2}{2} + \frac{7 x^{\frac{3}{2}}}{2}$

Step 5.2.15

Combine the numerators over the common denominator.

$8 x + \frac{14 x^{\frac{3}{2}} \cdot 2 + 7 x^{\frac{3}{2}}}{2}$

Step 5.2.16

Multiply $2$ by $14$ .

$8 x + \frac{28 x^{\frac{3}{2}} + 7 x^{\frac{3}{2}}}{2}$

Step 5.2.17

Add $28 x^{\frac{3}{2}}$ and $7 x^{\frac{3}{2}}$ .

$8 x + \frac{35 x^{\frac{3}{2}}}{2}$