Calculus Examples

$y = 3 x^{- \frac{3}{2}} + 2 x^{- \frac{1}{2}} + x^{3} - 2$

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [3 x^{- \frac{3}{2}}]$ .

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Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{3}{2}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $- 3$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

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

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

Step 2.9

Multiply $- 1$ by $3$ .

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

Step 2.10

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

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

Step 2.11

Multiply $3$ by $- 3$ .

$\frac{- 9 x^{- \frac{5}{2}}}{2} + \frac{d}{d x} [2 x^{- \frac{1}{2}}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 2.12

Move $x^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 9}{2 x^{\frac{5}{2}}} + \frac{d}{d x} [2 x^{- \frac{1}{2}}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 2.13

Move the negative in front of the fraction.

$- \frac{9}{2 x^{\frac{5}{2}}} + \frac{d}{d x} [2 x^{- \frac{1}{2}}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3

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

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Step 3.1

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

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 \frac{d}{d x} [x^{- \frac{1}{2}}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.2

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

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{1}{2} x^{- \frac{1}{2} - 1}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.3

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

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.4

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

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.5

Combine the numerators over the common denominator.

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{1}{2} x^{\frac{- 1 - 2}{2}}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.6.2

Subtract $2$ from $- 1$ .

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{1}{2} x^{\frac{- 3}{2}}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.7

Move the negative in front of the fraction.

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{1}{2} x^{- \frac{3}{2}}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.8

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

$- \frac{9}{2 x^{\frac{5}{2}}} + 2 (- \frac{x^{- \frac{3}{2}}}{2}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.9

Multiply $- 1$ by $2$ .

$- \frac{9}{2 x^{\frac{5}{2}}} - 2 \frac{x^{- \frac{3}{2}}}{2} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.10

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

$- \frac{9}{2 x^{\frac{5}{2}}} + \frac{- 2 x^{- \frac{3}{2}}}{2} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.11

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{9}{2 x^{\frac{5}{2}}} + \frac{- 2}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.12

Factor $2$ out of $- 2$ .

$- \frac{9}{2 x^{\frac{5}{2}}} + \frac{2 \cdot - 1}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$- \frac{9}{2 x^{\frac{5}{2}}} + \frac{2 \cdot - 1}{2 (x^{\frac{3}{2}})} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.13.2

Cancel the common factor.

$- \frac{9}{2 x^{\frac{5}{2}}} + \frac{2 \cdot - 1}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.13.3

Rewrite the expression.

$- \frac{9}{2 x^{\frac{5}{2}}} + \frac{- 1}{x^{\frac{3}{2}}} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 3.14

Move the negative in front of the fraction.

$- \frac{9}{2 x^{\frac{5}{2}}} - \frac{1}{x^{\frac{3}{2}}} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 4

Differentiate.

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Step 4.1

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

$- \frac{9}{2 x^{\frac{5}{2}}} - \frac{1}{x^{\frac{3}{2}}} + 3 x^{2} + \frac{d}{d x} [- 2]$

Step 4.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$- \frac{9}{2 x^{\frac{5}{2}}} - \frac{1}{x^{\frac{3}{2}}} + 3 x^{2} + 0$

Step 5

Simplify.

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Step 5.1

Add $- \frac{9}{2 x^{\frac{5}{2}}} - \frac{1}{x^{\frac{3}{2}}} + 3 x^{2}$ and $0$ .

$- \frac{9}{2 x^{\frac{5}{2}}} - \frac{1}{x^{\frac{3}{2}}} + 3 x^{2}$

Step 5.2

Reorder terms.

$3 x^{2} - \frac{9}{2 x^{\frac{5}{2}}} - \frac{1}{x^{\frac{3}{2}}}$