Calculus Examples

$g (x) = \sqrt{x} + {(x - 3)}^{3}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 3

Evaluate $\frac{d}{d x} [{(x - 3)}^{3}]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x - 3$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x - 3$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d u} [u^{3}] \frac{d}{d x} [x - 3]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$\frac{1}{2} x^{- \frac{1}{2}} + 3 u^{2} \frac{d}{d x} [x - 3]$

Step 3.1.3

Replace all occurrences of $u$ with $x - 3$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $0$ .

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

Step 3.6

Multiply $3$ by $1$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

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

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

Step 5.1.2

To write $3 {(x - 3)}^{2}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

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

Step 5.1.3

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

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

Step 5.1.4

Combine the numerators over the common denominator.

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

Step 5.1.5

Multiply $2$ by $3$ .

$\frac{1 + 6 {(x - 3)}^{2} x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 5.2

Reorder terms.

$\frac{6 x^{\frac{1}{2}} {(x - 3)}^{2} + 1}{2 x^{\frac{1}{2}}}$