Calculus Examples

$\frac{x^{0.5} - 2 x^{2} + 5 x^{1.5}}{2 x^{1.5}}$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{0.5} - 2 x^{2} + 5 x^{1.5}}{2 x^{1.5}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{x^{0.5} - 2 x^{2} + 5 x^{1.5}}{x^{1.5}}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{0.5} - 2 x^{2} + 5 x^{1.5}}{x^{1.5}}]$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $x^{0.5}$ out of $x^{0.5} - 2 x^{2} + 5 x^{1.5}$ .

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Step 2.1.1

Multiply by $1$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{0.5} \cdot 1 - 2 x^{2} + 5 x^{1.5}}{x^{1.5}}]$

Step 2.1.2

Factor $x^{0.5}$ out of $- 2 x^{2}$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{0.5} \cdot 1 + x^{0.5} (- 2 x^{1.5}) + 5 x^{1.5}}{x^{1.5}}]$

Step 2.1.3

Factor $x^{0.5}$ out of $5 x^{1.5}$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{0.5} \cdot 1 + x^{0.5} (- 2 x^{1.5}) + x^{0.5} (5 x)}{x^{1.5}}]$

Step 2.1.4

Factor $x^{0.5}$ out of $x^{0.5} \cdot 1 + x^{0.5} (- 2 x^{1.5})$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{0.5} \cdot (1 - 2 x^{1.5}) + x^{0.5} (5 x)}{x^{1.5}}]$

Step 2.1.5

Factor $x^{0.5}$ out of $x^{0.5} \cdot (1 - 2 x^{1.5}) + x^{0.5} (5 x)$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{0.5} (1 - 2 x^{1.5} + 5 x)}{x^{1.5}}]$

Step 2.2

Move $x^{0.5}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{2} \frac{d}{d x} [\frac{1 - 2 x^{1.5} + 5 x}{x^{1.5} x^{- 0.5}}]$

Step 3

Simplify the denominator.

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Step 3.1

Multiply $x^{1.5}$ by $x^{- 0.5}$ by adding the exponents.

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Step 3.1.1

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

$\frac{1}{2} \frac{d}{d x} [\frac{1 - 2 x^{1.5} + 5 x}{x^{1.5 - 0.5}}]$

Step 3.1.2

Subtract $0.5$ from $1.5$ .

$\frac{1}{2} \frac{d}{d x} [\frac{1 - 2 x^{1.5} + 5 x}{x^{1}}]$

Step 3.2

Simplify $x^{1}$ .

$\frac{1}{2} \frac{d}{d x} [\frac{1 - 2 x^{1.5} + 5 x}{x}]$

Step 4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 - 2 x^{1.5} + 5 x$ and $g (x) = x$ .

$\frac{1}{2} \cdot \frac{x \frac{d}{d x} [1 - 2 x^{1.5} + 5 x] - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $1 - 2 x^{1.5} + 5 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x^{1.5}] + \frac{d}{d x} [5 x]$ .

$\frac{1}{2} \cdot \frac{x (\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x^{1.5}] + \frac{d}{d x} [5 x]) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{2} \cdot \frac{x (0 + \frac{d}{d x} [- 2 x^{1.5}] + \frac{d}{d x} [5 x]) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.3

Add $0$ and $\frac{d}{d x} [- 2 x^{1.5}]$ .

$\frac{1}{2} \cdot \frac{x (\frac{d}{d x} [- 2 x^{1.5}] + \frac{d}{d x} [5 x]) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{1.5}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{1.5}]$ .

$\frac{1}{2} \cdot \frac{x (- 2 \frac{d}{d x} [x^{1.5}] + \frac{d}{d x} [5 x]) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1.5$ .

$\frac{1}{2} \cdot \frac{x (- 2 (1.5 x^{0.5}) + \frac{d}{d x} [5 x]) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.6

Multiply $1.5$ by $- 2$ .

$\frac{1}{2} \cdot \frac{x (- 3 x^{0.5} + \frac{d}{d x} [5 x]) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.7

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

$\frac{1}{2} \cdot \frac{x (- 3 x^{0.5} + 5 \frac{d}{d x} [x]) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.8

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

$\frac{1}{2} \cdot \frac{x (- 3 x^{0.5} + 5 \cdot 1) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.9

Multiply $5$ by $1$ .

$\frac{1}{2} \cdot \frac{x (- 3 x^{0.5} + 5) - (1 - 2 x^{1.5} + 5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5.10

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

$\frac{1}{2} \cdot \frac{x (- 3 x^{0.5} + 5) - (1 - 2 x^{1.5} + 5 x) \cdot 1}{x^{2}}$

Step 5.11

Combine fractions.

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Step 5.11.1

Multiply $- 1$ by $1$ .

$\frac{1}{2} \cdot \frac{x (- 3 x^{0.5} + 5) - (1 - 2 x^{1.5} + 5 x)}{x^{2}}$

Step 5.11.2

Multiply $\frac{1}{2}$ by $\frac{x (- 3 x^{0.5} + 5) - (1 - 2 x^{1.5} + 5 x)}{x^{2}}$ .

$\frac{x (- 3 x^{0.5} + 5) - (1 - 2 x^{1.5} + 5 x)}{2 x^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{x (- 3 x^{0.5}) + x \cdot 5 - (1 - 2 x^{1.5} + 5 x)}{2 x^{2}}$

Step 6.2

Apply the distributive property.

$\frac{x (- 3 x^{0.5}) + x \cdot 5 - 1 \cdot 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 3 x \cdot x^{0.5} + x \cdot 5 - 1 \cdot 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3.1.2

Multiply $x$ by $x^{0.5}$ by adding the exponents.

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Step 6.3.1.2.1

Move $x^{0.5}$ .

$\frac{- 3 (x^{0.5} x) + x \cdot 5 - 1 \cdot 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3.1.2.2

Multiply $x^{0.5}$ by $x$ .

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Step 6.3.1.2.2.1

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

$\frac{- 3 (x^{0.5} x^{1}) + x \cdot 5 - 1 \cdot 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3.1.2.2.2

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

$\frac{- 3 x^{0.5 + 1} + x \cdot 5 - 1 \cdot 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3.1.2.3

Add $0.5$ and $1$ .

$\frac{- 3 x^{1.5} + x \cdot 5 - 1 \cdot 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3.1.3

Move $5$ to the left of $x$ .

$\frac{- 3 x^{1.5} + 5 \cdot x - 1 \cdot 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3.1.4

Multiply $- 1$ by $1$ .

$\frac{- 3 x^{1.5} + 5 x - 1 - (- 2 x^{1.5}) - (5 x)}{2 x^{2}}$

Step 6.3.1.5

Multiply $- 2$ by $- 1$ .

$\frac{- 3 x^{1.5} + 5 x - 1 + 2 x^{1.5} - (5 x)}{2 x^{2}}$

Step 6.3.1.6

Multiply $5$ by $- 1$ .

$\frac{- 3 x^{1.5} + 5 x - 1 + 2 x^{1.5} - 5 x}{2 x^{2}}$

Step 6.3.2

Combine the opposite terms in $- 3 x^{1.5} + 5 x - 1 + 2 x^{1.5} - 5 x$ .

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Step 6.3.2.1

Subtract $5 x$ from $5 x$ .

$\frac{- 3 x^{1.5} - 1 + 2 x^{1.5} + 0}{2 x^{2}}$

Step 6.3.2.2

Add $- 3 x^{1.5} - 1 + 2 x^{1.5}$ and $0$ .

$\frac{- 3 x^{1.5} - 1 + 2 x^{1.5}}{2 x^{2}}$

Step 6.3.3

Add $- 3 x^{1.5}$ and $2 x^{1.5}$ .

$\frac{- 1 x^{1.5} - 1}{2 x^{2}}$

Step 6.4

Factor $- 1$ out of $- 1 x^{1.5}$ .

$\frac{- (1 x^{1.5}) - 1}{2 x^{2}}$

Step 6.5

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- (1 x^{1.5}) - 1 (1)}{2 x^{2}}$

Step 6.6

Factor $- 1$ out of $- (1 x^{1.5}) - 1 (1)$ .

$\frac{- (1 x^{1.5} + 1)}{2 x^{2}}$

Step 6.7

Rewrite $- (1 x^{1.5} + 1)$ as $- 1 (1 x^{1.5} + 1)$ .

$\frac{- 1 (1 x^{1.5} + 1)}{2 x^{2}}$

Step 6.8

Move the negative in front of the fraction.

$- \frac{1 x^{1.5} + 1}{2 x^{2}}$

Step 6.9

Multiply $x^{1.5}$ by $1$ .

$- \frac{x^{1.5} + 1}{2 x^{2}}$