Calculus Examples

$y = x^{2} \sqrt{x} - \frac{5}{x}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$\frac{d}{d x} [x^{2 \cdot \frac{2}{2} + \frac{1}{2}}] + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.2.3

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

$\frac{d}{d x} [x^{\frac{2 \cdot 2}{2} + \frac{1}{2}}] + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{2 \cdot 2 + 1}{2}}] + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $2$ by $2$ .

$\frac{d}{d x} [x^{\frac{4 + 1}{2}}] + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.2.5.2

Add $4$ and $1$ .

$\frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

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

Step 2.4

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

$\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.5

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

$\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.6

Combine the numerators over the common denominator.

$\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}} + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$\frac{5}{2} x^{\frac{5 - 2}{2}} + \frac{d}{d x} [- \frac{5}{x}]$

Step 2.7.2

Subtract $2$ from $5$ .

$\frac{5}{2} x^{\frac{3}{2}} + \frac{d}{d x} [- \frac{5}{x}]$

Step 3

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

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Step 3.1

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

$\frac{5}{2} x^{\frac{3}{2}} - 5 \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

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

$\frac{5}{2} x^{\frac{3}{2}} - 5 \frac{d}{d x} [x^{- 1}]$

Step 3.3

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

$\frac{5}{2} x^{\frac{3}{2}} - 5 (- x^{- 2})$

Step 3.4

Multiply $- 1$ by $- 5$ .

$\frac{5}{2} x^{\frac{3}{2}} + 5 x^{- 2}$

Step 4

Simplify.

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Step 4.1

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

$\frac{5}{2} x^{\frac{3}{2}} + 5 \frac{1}{x^{2}}$

Step 4.2

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

$\frac{5}{2} x^{\frac{3}{2}} + \frac{5}{x^{2}}$

Step 4.3

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$\frac{5 x^{\frac{3}{2}}}{2} + \frac{5}{x^{2}}$