Calculus Examples

$f (x) = \sqrt{3 p} \cdot \sqrt[3]{x^{4}}$

Step 1

Factor out $x^{3}$ .

$\frac{d}{d p} [\sqrt{3 p} \cdot \sqrt[3]{x^{3} x}]$

Step 2

Pull terms out from under the radical.

$\frac{d}{d p} [\sqrt{3 p} \cdot (x \sqrt[3]{x})]$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{d}{d p} [\sqrt{3 p} \cdot (x \cdot x^{\frac{1}{3}})]$

Step 4

Multiply $x$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 4.1

Multiply $x$ by $x^{\frac{1}{3}}$ .

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Step 4.1.1

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

$\frac{d}{d p} [\sqrt{3 p} \cdot (x^{1} x^{\frac{1}{3}})]$

Step 4.1.2

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

$\frac{d}{d p} [\sqrt{3 p} \cdot x^{1 + \frac{1}{3}}]$

Step 4.2

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

$\frac{d}{d p} [\sqrt{3 p} \cdot x^{\frac{3}{3} + \frac{1}{3}}]$

Step 4.3

Combine the numerators over the common denominator.

$\frac{d}{d p} [\sqrt{3 p} \cdot x^{\frac{3 + 1}{3}}]$

Step 4.4

Add $3$ and $1$ .

$\frac{d}{d p} [\sqrt{3 p} \cdot x^{\frac{4}{3}}]$

Step 5

Simplify with factoring out.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 p}$ as ${(3 p)}^{\frac{1}{2}}$ .

$\frac{d}{d p} [{(3 p)}^{\frac{1}{2}} x^{\frac{4}{3}}]$

Step 5.2

Factor $3$ out of $3 p$ .

$\frac{d}{d p} [{(3 (p))}^{\frac{1}{2}} x^{\frac{4}{3}}]$

Step 5.3

Apply the product rule to $3 (p)$ .

$\frac{d}{d p} [3^{\frac{1}{2}} p^{\frac{1}{2}} x^{\frac{4}{3}}]$

Step 6

Since $3^{\frac{1}{2}} x^{\frac{4}{3}}$ is constant with respect to $p$ , the derivative of $3^{\frac{1}{2}} p^{\frac{1}{2}} x^{\frac{4}{3}}$ with respect to $p$ is $3^{\frac{1}{2}} x^{\frac{4}{3}} \frac{d}{d p} [p^{\frac{1}{2}}]$ .

$3^{\frac{1}{2}} x^{\frac{4}{3}} \frac{d}{d p} [p^{\frac{1}{2}}]$

Step 7

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

$3^{\frac{1}{2}} x^{\frac{4}{3}} (\frac{1}{2} p^{\frac{1}{2} - 1})$

Step 8

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

$3^{\frac{1}{2}} x^{\frac{4}{3}} (\frac{1}{2} p^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 9

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

$3^{\frac{1}{2}} x^{\frac{4}{3}} (\frac{1}{2} p^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 10

Combine the numerators over the common denominator.

$3^{\frac{1}{2}} x^{\frac{4}{3}} (\frac{1}{2} p^{\frac{1 - 1 \cdot 2}{2}})$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $- 1$ by $2$ .

$3^{\frac{1}{2}} x^{\frac{4}{3}} (\frac{1}{2} p^{\frac{1 - 2}{2}})$

Step 11.2

Subtract $2$ from $1$ .

$3^{\frac{1}{2}} x^{\frac{4}{3}} (\frac{1}{2} p^{\frac{- 1}{2}})$

Step 12

Move the negative in front of the fraction.

$3^{\frac{1}{2}} x^{\frac{4}{3}} (\frac{1}{2} p^{- \frac{1}{2}})$

Step 13

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

$3^{\frac{1}{2}} x^{\frac{4}{3}} \frac{p^{- \frac{1}{2}}}{2}$

Step 14

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

$x^{\frac{4}{3}} \frac{3^{\frac{1}{2}} p^{- \frac{1}{2}}}{2}$

Step 15

Combine $x^{\frac{4}{3}}$ and $\frac{3^{\frac{1}{2}} p^{- \frac{1}{2}}}{2}$ .

$\frac{x^{\frac{4}{3}} (3^{\frac{1}{2}} p^{- \frac{1}{2}})}{2}$

Step 16

Simplify the expression.

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Step 16.1

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

$\frac{x^{\frac{4}{3}} \cdot 3^{\frac{1}{2}}}{2 p^{\frac{1}{2}}}$

Step 16.2

Reorder terms.

$\frac{3^{\frac{1}{2}} x^{\frac{4}{3}}}{2 p^{\frac{1}{2}}}$