Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 2 x - 3$ and $g (x) = x^{\frac{3}{2}}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $3$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Multiply $2$ by $1$ .

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

Step 12

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(2 x - 3) \frac{3 x^{\frac{1}{2}}}{2} + x^{\frac{3}{2}} (2 + 0)$

Step 13

Simplify the expression.

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Step 13.1

Add $2$ and $0$ .

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

Step 13.2

Move $2$ to the left of $x^{\frac{3}{2}}$ .

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

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Combine terms.

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Step 14.2.1

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

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

Step 14.2.2

Multiply $3$ by $2$ .

$x \frac{6 x^{\frac{1}{2}}}{2} - 3 \frac{3 x^{\frac{1}{2}}}{2} + 2 x^{\frac{3}{2}}$

Step 14.2.3

Combine $x$ and $\frac{6 x^{\frac{1}{2}}}{2}$ .

$\frac{x (6 x^{\frac{1}{2}})}{2} - 3 \frac{3 x^{\frac{1}{2}}}{2} + 2 x^{\frac{3}{2}}$

Step 14.2.4

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

$\frac{6 (x^{1} x^{\frac{1}{2}})}{2} - 3 \frac{3 x^{\frac{1}{2}}}{2} + 2 x^{\frac{3}{2}}$

Step 14.2.5

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

$\frac{6 x^{1 + \frac{1}{2}}}{2} - 3 \frac{3 x^{\frac{1}{2}}}{2} + 2 x^{\frac{3}{2}}$

Step 14.2.6

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

$\frac{6 x^{\frac{2}{2} + \frac{1}{2}}}{2} - 3 \frac{3 x^{\frac{1}{2}}}{2} + 2 x^{\frac{3}{2}}$

Step 14.2.7

Combine the numerators over the common denominator.

$\frac{6 x^{\frac{2 + 1}{2}}}{2} - 3 \frac{3 x^{\frac{1}{2}}}{2} + 2 x^{\frac{3}{2}}$

Step 14.2.8

Add $2$ and $1$ .

$\frac{6 x^{\frac{3}{2}}}{2} - 3 \frac{3 x^{\frac{1}{2}}}{2} + 2 x^{\frac{3}{2}}$

Step 14.2.9

Factor $2$ out of $6 x^{\frac{3}{2}}$ .

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

Step 14.2.10

Cancel the common factors.

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Step 14.2.10.1

Factor $2$ out of $2$ .

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

Step 14.2.10.2

Cancel the common factor.

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

Step 14.2.10.3

Rewrite the expression.

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

Step 14.2.10.4

Divide $3 x^{\frac{3}{2}}$ by $1$ .

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

Step 14.2.11

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

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

Step 14.2.12

Multiply $3$ by $- 3$ .

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

Step 14.2.13

Move the negative in front of the fraction.

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

Step 14.2.14

Add $3 x^{\frac{3}{2}}$ and $2 x^{\frac{3}{2}}$ .

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