Calculus Examples

$y = (x + 1) (7 \sqrt{x} + 2)$

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 $f (x) = x + 1$ and $g (x) = 7 \sqrt{x} + 2$ .

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

Step 2

By the Sum Rule, the derivative of $7 \sqrt{x} + 2$ with respect to $x$ is $\frac{d}{d x} [7 \sqrt{x}] + \frac{d}{d x} [2]$ .

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

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.7.2

Subtract $2$ from $1$ .

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

Step 3.8

Move the negative in front of the fraction.

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 4

Differentiate.

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Step 4.1

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

$(x + 1) (\frac{7}{2 x^{\frac{1}{2}}} + 0) + (7 \sqrt{x} + 2) \frac{d}{d x} [x + 1]$

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 4.6

Add $1$ and $0$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

$x \frac{7}{2 x^{\frac{1}{2}}} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3

Combine terms.

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Step 5.3.1

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

$\frac{x \cdot 7}{2 x^{\frac{1}{2}}} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.2

Move $7$ to the left of $x$ .

$\frac{7 \cdot x}{2 x^{\frac{1}{2}}} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.3

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

$\frac{7 x \cdot x^{- \frac{1}{2}}}{2} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.4

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

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Step 5.3.4.1

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

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

Step 5.3.4.2

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

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Step 5.3.4.2.1

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

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

Step 5.3.4.2.2

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

$\frac{7 x^{- \frac{1}{2} + 1}}{2} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.4.3

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

$\frac{7 x^{- \frac{1}{2} + \frac{2}{2}}}{2} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.4.4

Combine the numerators over the common denominator.

$\frac{7 x^{\frac{- 1 + 2}{2}}}{2} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.4.5

Add $- 1$ and $2$ .

$\frac{7 x^{\frac{1}{2}}}{2} + 1 \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.5

Multiply $\frac{7}{2 x^{\frac{1}{2}}}$ by $1$ .

$\frac{7 x^{\frac{1}{2}}}{2} + \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} \cdot 1 + 2 \cdot 1$

Step 5.3.6

Multiply $7$ by $1$ .

$\frac{7 x^{\frac{1}{2}}}{2} + \frac{7}{2 x^{\frac{1}{2}}} + 7 \sqrt{x} + 2 \cdot 1$

Step 5.3.7

Multiply $2$ by $1$ .

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

Step 5.4

Reorder terms.

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