Calculus Examples

$2 x + 25 x^{\frac{2}{25}}$

Step 1

By the Sum Rule, the derivative of $2 x + 25 x^{\frac{2}{25}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [25 x^{\frac{2}{25}}]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [25 x^{\frac{2}{25}}]$

Step 2

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

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Step 2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [25 x^{\frac{2}{25}}]$

Step 2.2

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

$2 \cdot 1 + \frac{d}{d x} [25 x^{\frac{2}{25}}]$

Step 2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [25 x^{\frac{2}{25}}]$

Step 3

Evaluate $\frac{d}{d x} [25 x^{\frac{2}{25}}]$ .

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Step 3.1

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{\frac{2}{25}}$ with respect to $x$ is $25 \frac{d}{d x} [x^{\frac{2}{25}}]$ .

$2 + 25 \frac{d}{d x} [x^{\frac{2}{25}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{25}$ .

$2 + 25 (\frac{2}{25} x^{\frac{2}{25} - 1})$

Step 3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$2 + 25 (\frac{2}{25} x^{\frac{2}{25} - 1 \cdot \frac{25}{25}})$

Step 3.4

Combine $- 1$ and $\frac{25}{25}$ .

$2 + 25 (\frac{2}{25} x^{\frac{2}{25} + \frac{- 1 \cdot 25}{25}})$

Step 3.5

Combine the numerators over the common denominator.

$2 + 25 (\frac{2}{25} x^{\frac{2 - 1 \cdot 25}{25}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $25$ .

$2 + 25 (\frac{2}{25} x^{\frac{2 - 25}{25}})$

Step 3.6.2

Subtract $25$ from $2$ .

$2 + 25 (\frac{2}{25} x^{\frac{- 23}{25}})$

Step 3.7

Move the negative in front of the fraction.

$2 + 25 (\frac{2}{25} x^{- \frac{23}{25}})$

Step 3.8

Combine $\frac{2}{25}$ and $x^{- \frac{23}{25}}$ .

$2 + 25 \frac{2 x^{- \frac{23}{25}}}{25}$

Step 3.9

Combine $25$ and $\frac{2 x^{- \frac{23}{25}}}{25}$ .

$2 + \frac{25 (2 x^{- \frac{23}{25}})}{25}$

Step 3.10

Multiply $2$ by $25$ .

$2 + \frac{50 x^{- \frac{23}{25}}}{25}$

Step 3.11

Move $x^{- \frac{23}{25}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 + \frac{50}{25 x^{\frac{23}{25}}}$

Step 3.12

Factor $25$ out of $50$ .

$2 + \frac{25 \cdot 2}{25 x^{\frac{23}{25}}}$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $25$ out of $25 x^{\frac{23}{25}}$ .

$2 + \frac{25 \cdot 2}{25 (x^{\frac{23}{25}})}$

Step 3.13.2

Cancel the common factor.

$2 + \frac{25 \cdot 2}{25 x^{\frac{23}{25}}}$

Step 3.13.3

Rewrite the expression.

$2 + \frac{2}{x^{\frac{23}{25}}}$