Algebra Examples

$\sqrt[n]{x^{2 n + 1}}$

Step 1

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

$\frac{d}{d n} [x^{\frac{2 n + 1}{n}}]$

Step 2

Differentiate using the Generalized Power Rule which states that $\frac{d}{d n} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = x$ and $g = \frac{2 n + 1}{n}$ .

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

Step 3

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

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

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

Step 4.2

Simplify terms.

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Subtract $n$ from $2 n$ .

$\frac{d}{d n} [x] (\frac{2 n + 1}{n} x^{\frac{n + 1}{n}}) + \frac{d}{d n} [\frac{2 n + 1}{n}] (x^{\frac{2 n + 1}{n}} \ln (x))$

Step 4.2.3

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

$\frac{d}{d n} [x] \frac{(2 n + 1) x^{\frac{n + 1}{n}}}{n} + \frac{d}{d n} [\frac{2 n + 1}{n}] (x^{\frac{2 n + 1}{n}} \ln (x))$

Step 4.3

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

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Multiply $0$ by $\frac{(2 n + 1) x^{\frac{n + 1}{n}}}{n}$ .

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

Step 4.4.2

Add $0$ and $\frac{d}{d n} [\frac{2 n + 1}{n}] (x^{\frac{2 n + 1}{n}} \ln (x))$ .

$\frac{d}{d n} [\frac{2 n + 1}{n}] (x^{\frac{2 n + 1}{n}} \ln (x))$

Step 5

Differentiate using the Quotient Rule which states that $\frac{d}{d n} [\frac{f (n)}{g (n)}]$ is $\frac{g (n) \frac{d}{d n} [f (n)] - f (n) \frac{d}{d n} [g (n)]}{{g (n)}^{2}}$ where $f (n) = 2 n + 1$ and $g (n) = n$ .

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

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $2 n + 1$ with respect to $n$ is $\frac{d}{d n} [2 n] + \frac{d}{d n} [1]$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Multiply $2$ by $1$ .

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

Step 6.5

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

$\frac{n (2 + 0) - (2 n + 1) \frac{d}{d n} [n]}{n^{2}} (x^{\frac{2 n + 1}{n}} \ln (x))$

Step 6.6

Simplify the expression.

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Step 6.6.1

Add $2$ and $0$ .

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

Step 6.6.2

Move $2$ to the left of $n$ .

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

Step 6.7

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

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

Step 6.8

Combine fractions.

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Step 6.8.1

Multiply $- 1$ by $1$ .

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

Step 6.8.2

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

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

Step 6.8.3

Combine $\frac{x^{\frac{2 n + 1}{n}} (2 n - (2 n + 1))}{n^{2}}$ and $\ln (x)$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Simplify the numerator.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $2$ by $- 1$ .

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

Step 7.2.1.2

Multiply $- 1$ by $1$ .

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

Step 7.2.2

Subtract $2 n$ from $2 n$ .

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

Step 7.2.3

Subtract $1$ from $0$ .

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

Step 7.2.4

Move $- 1$ to the left of $x^{\frac{2 n + 1}{n}}$ .

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

Step 7.2.5

Rewrite $- 1 x^{\frac{2 n + 1}{n}}$ as $- x^{\frac{2 n + 1}{n}}$ .

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

Step 7.3

Move the negative in front of the fraction.

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