Calculus Examples

$y = 9 x^{\frac{5}{2}} - 7 x^{\frac{3}{2}}$ , $x = 4$

Step 1

By the Sum Rule, the derivative of $9 x^{\frac{5}{2}} - 7 x^{\frac{3}{2}}$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{5}{2}}] + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$ .

$\frac{d}{d x} [9 x^{\frac{5}{2}}] + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2

Evaluate $\frac{d}{d x} [9 x^{\frac{5}{2}}]$ .

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Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{5}{2}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$9 \frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.2

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

$9 (\frac{5}{2} x^{\frac{5}{2} - 1}) + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.3

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

$9 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.4

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

$9 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.5

Combine the numerators over the common denominator.

$9 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$9 (\frac{5}{2} x^{\frac{5 - 2}{2}}) + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.6.2

Subtract $2$ from $5$ .

$9 (\frac{5}{2} x^{\frac{3}{2}}) + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.7

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

$9 \frac{5 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.8

Combine $9$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$\frac{9 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 2.9

Multiply $5$ by $9$ .

$\frac{45 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- 7 x^{\frac{3}{2}}]$

Step 3

Evaluate $\frac{d}{d x} [- 7 x^{\frac{3}{2}}]$ .

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Step 3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{\frac{3}{2}}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$\frac{45 x^{\frac{3}{2}}}{2} - 7 \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 3.2

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

$\frac{45 x^{\frac{3}{2}}}{2} - 7 (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 3.3

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

$\frac{45 x^{\frac{3}{2}}}{2} - 7 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 3.4

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

$\frac{45 x^{\frac{3}{2}}}{2} - 7 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.5

Combine the numerators over the common denominator.

$\frac{45 x^{\frac{3}{2}}}{2} - 7 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$\frac{45 x^{\frac{3}{2}}}{2} - 7 (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 3.6.2

Subtract $2$ from $3$ .

$\frac{45 x^{\frac{3}{2}}}{2} - 7 (\frac{3}{2} x^{\frac{1}{2}})$

Step 3.7

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

$\frac{45 x^{\frac{3}{2}}}{2} - 7 \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.8

Combine $- 7$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{45 x^{\frac{3}{2}}}{2} + \frac{- 7 (3 x^{\frac{1}{2}})}{2}$

Step 3.9

Multiply $3$ by $- 7$ .

$\frac{45 x^{\frac{3}{2}}}{2} + \frac{- 21 x^{\frac{1}{2}}}{2}$

Step 3.10

Move the negative in front of the fraction.

$\frac{45 x^{\frac{3}{2}}}{2} - \frac{21 x^{\frac{1}{2}}}{2}$

Step 4

Evaluate the derivative at $x = 4$ .

$\frac{45 {(4)}^{\frac{3}{2}}}{2} - \frac{21 {(4)}^{\frac{1}{2}}}{2}$

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{45 {(4)}^{\frac{3}{2}} - 21 {(4)}^{\frac{1}{2}}}{2}$

Step 5.2

Simplify each term.

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Step 5.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{45 {(2^{2})}^{\frac{3}{2}} - 21 {(4)}^{\frac{1}{2}}}{2}$

Step 5.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{45 \cdot 2^{2 (\frac{3}{2})} - 21 {(4)}^{\frac{1}{2}}}{2}$

Step 5.2.3

Cancel the common factor of $2$ .

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Step 5.2.3.1

Cancel the common factor.

$\frac{45 \cdot 2^{2 (\frac{3}{2})} - 21 {(4)}^{\frac{1}{2}}}{2}$

Step 5.2.3.2

Rewrite the expression.

$\frac{45 \cdot 2^{3} - 21 {(4)}^{\frac{1}{2}}}{2}$

Step 5.2.4

Raise $2$ to the power of $3$ .

$\frac{45 \cdot 8 - 21 {(4)}^{\frac{1}{2}}}{2}$

Step 5.2.5

Multiply $45$ by $8$ .

$\frac{360 - 21 {(4)}^{\frac{1}{2}}}{2}$

Step 5.2.6

Rewrite $4$ as $2^{2}$ .

$\frac{360 - 21 {(2^{2})}^{\frac{1}{2}}}{2}$

Step 5.2.7

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{360 - 21 \cdot 2^{2 (\frac{1}{2})}}{2}$

Step 5.2.8

Cancel the common factor of $2$ .

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Step 5.2.8.1

Cancel the common factor.

$\frac{360 - 21 \cdot 2^{2 (\frac{1}{2})}}{2}$

Step 5.2.8.2

Rewrite the expression.

$\frac{360 - 21 \cdot 2^{1}}{2}$

Step 5.2.9

Evaluate the exponent.

$\frac{360 - 21 \cdot 2}{2}$

Step 5.2.10

Multiply $- 21$ by $2$ .

$\frac{360 - 42}{2}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Subtract $42$ from $360$ .

$\frac{318}{2}$

Step 5.3.2

Divide $318$ by $2$ .

$159$