Calculus Examples

$f (x) = (3 x - 5) (\sqrt{x} + 3)$

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 $f (x) = 3 x - 5$ and $g (x) = \sqrt{x} + 3$ .

$(3 x - 5) \frac{d}{d x} [\sqrt{x} + 3] + (\sqrt{x} + 3) \frac{d}{d x} [3 x - 5]$

Step 2

By the Sum Rule, the derivative of $\sqrt{x} + 3$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [3]$ .

$(3 x - 5) (\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [3]) + (\sqrt{x} + 3) \frac{d}{d x} [3 x - 5]$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $1$ .

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

Step 3.7

Move the negative in front of the fraction.

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

Step 4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$(3 x - 5) (\frac{1}{2} x^{- \frac{1}{2}} + 0) + (\sqrt{x} + 3) \frac{d}{d x} [3 x - 5]$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(3 x - 5) (\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + 0) + (\sqrt{x} + 3) \frac{d}{d x} [3 x - 5]$

Step 5.2

Combine terms.

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Step 5.2.1

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

$(3 x - 5) (\frac{1}{2 x^{\frac{1}{2}}} + 0) + (\sqrt{x} + 3) \frac{d}{d x} [3 x - 5]$

Step 5.2.2

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

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

Step 6

By the Sum Rule, the derivative of $3 x - 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

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

Step 7

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 7.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 7.2

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

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

Step 7.3

Multiply $3$ by $1$ .

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

Step 8

Differentiate using the Constant Rule.

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Step 8.1

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

$(3 x - 5) \frac{1}{2 x^{\frac{1}{2}}} + (\sqrt{x} + 3) (3 + 0)$

Step 8.2

Add $3$ and $0$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

$3 x \frac{1}{2 x^{\frac{1}{2}}} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3

Combine terms.

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Step 9.3.1

Combine $3$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$x \frac{3}{2 x^{\frac{1}{2}}} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.2

Combine $x$ and $\frac{3}{2 x^{\frac{1}{2}}}$ .

$\frac{x \cdot 3}{2 x^{\frac{1}{2}}} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.3

Move $3$ to the left of $x$ .

$\frac{3 \cdot x}{2 x^{\frac{1}{2}}} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.4

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

$\frac{3 x \cdot x^{- \frac{1}{2}}}{2} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.5

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

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Step 9.3.5.1

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

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

Step 9.3.5.2

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

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Step 9.3.5.2.1

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

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

Step 9.3.5.2.2

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

$\frac{3 x^{- \frac{1}{2} + 1}}{2} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.5.3

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

$\frac{3 x^{- \frac{1}{2} + \frac{2}{2}}}{2} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.5.4

Combine the numerators over the common denominator.

$\frac{3 x^{\frac{- 1 + 2}{2}}}{2} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.5.5

Add $- 1$ and $2$ .

$\frac{3 x^{\frac{1}{2}}}{2} - 5 \frac{1}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.6

Combine $- 5$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{3 x^{\frac{1}{2}}}{2} + \frac{- 5}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.7

Move the negative in front of the fraction.

$\frac{3 x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}} + \sqrt{x} \cdot 3 + 3 \cdot 3$

Step 9.3.8

Move $3$ to the left of $\sqrt{x}$ .

$\frac{3 x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}} + 3 \cdot \sqrt{x} + 3 \cdot 3$

Step 9.3.9

Multiply $3$ by $3$ .

$\frac{3 x^{\frac{1}{2}}}{2} - \frac{5}{2 x^{\frac{1}{2}}} + 3 \sqrt{x} + 9$

Step 9.4

Reorder terms.

$\frac{3 x^{\frac{1}{2}}}{2} + 9 - \frac{5}{2 x^{\frac{1}{2}}} + 3 \sqrt{x}$