Calculus Examples

$f (x) = 5 \sqrt{x} + 6 x^{3} - \frac{1}{2} x^{2} + \sqrt{π}$

Step 1

By the Sum Rule, the derivative of $5 \sqrt{x} + 6 x^{3} - \frac{1}{2} x^{2} + \sqrt{π}$ with respect to $x$ is $\frac{d}{d x} [5 \sqrt{x}] + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$ .

$\frac{d}{d x} [5 \sqrt{x}] + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2

Evaluate $\frac{d}{d x} [5 \sqrt{x}]$ .

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Step 2.1

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

$\frac{d}{d x} [5 x^{\frac{1}{2}}] + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{1}{2}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$5 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.3

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

$5 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.4

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

$5 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.5

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

$5 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.6

Combine the numerators over the common denominator.

$5 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$5 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.7.2

Subtract $2$ from $1$ .

$5 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.8

Move the negative in front of the fraction.

$5 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.9

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

$5 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.10

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

$\frac{5 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 2.11

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

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 3

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 3.2

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

$\frac{5}{2 x^{\frac{1}{2}}} + 6 (3 x^{2}) + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 3.3

Multiply $3$ by $6$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 4

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{2}]$ .

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Step 4.1

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

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} - \frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\sqrt{π}]$

Step 4.2

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

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} - \frac{1}{2} (2 x) + \frac{d}{d x} [\sqrt{π}]$

Step 4.3

Multiply $2$ by $- 1$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} - 2 (\frac{1}{2}) x + \frac{d}{d x} [\sqrt{π}]$

Step 4.4

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

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} + \frac{- 2}{2} x + \frac{d}{d x} [\sqrt{π}]$

Step 4.5

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

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} + \frac{- 2 x}{2} + \frac{d}{d x} [\sqrt{π}]$

Step 4.6

Cancel the common factor of $- 2$ and $2$ .

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Step 4.6.1

Factor $2$ out of $- 2 x$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} + \frac{2 (- x)}{2} + \frac{d}{d x} [\sqrt{π}]$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $2$ out of $2$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} + \frac{2 (- x)}{2 (1)} + \frac{d}{d x} [\sqrt{π}]$

Step 4.6.2.2

Cancel the common factor.

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} + \frac{2 (- x)}{2 \cdot 1} + \frac{d}{d x} [\sqrt{π}]$

Step 4.6.2.3

Rewrite the expression.

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} + \frac{- x}{1} + \frac{d}{d x} [\sqrt{π}]$

Step 4.6.2.4

Divide $- x$ by $1$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} - x + \frac{d}{d x} [\sqrt{π}]$

Step 5

Since $\sqrt{π}$ is constant with respect to $x$ , the derivative of $\sqrt{π}$ with respect to $x$ is $0$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} - x + 0$

Step 6

Simplify.

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Step 6.1

Add $\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} - x$ and $0$ .

$\frac{5}{2 x^{\frac{1}{2}}} + 18 x^{2} - x$

Step 6.2

Reorder terms.

$18 x^{2} - x + \frac{5}{2 x^{\frac{1}{2}}}$