Calculus Examples

$y = \frac{5 + 6 x - 5 x^{2}}{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{5 + 6 x - 5 x^{2}}{x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = 5 + 6 x - 5 x^{2}$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [5 + 6 x - 5 x^{2}] - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $5 + 6 x - 5 x^{2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 5 x^{2}]$ .

$\frac{x (\frac{d}{d x} [5] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 5 x^{2}]) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{x (0 + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 5 x^{2}]) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [6 x]$ .

$\frac{x (\frac{d}{d x} [6 x] + \frac{d}{d x} [- 5 x^{2}]) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.4

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{x (6 \frac{d}{d x} [x] + \frac{d}{d x} [- 5 x^{2}]) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.5

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

$\frac{x (6 \cdot 1 + \frac{d}{d x} [- 5 x^{2}]) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.6

Multiply $6$ by $1$ .

$\frac{x (6 + \frac{d}{d x} [- 5 x^{2}]) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.7

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

$\frac{x (6 - 5 \frac{d}{d x} [x^{2}]) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.8

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

$\frac{x (6 - 5 (2 x)) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.9

Multiply $2$ by $- 5$ .

$\frac{x (6 - 10 x) - (5 + 6 x - 5 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.10

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

$\frac{x (6 - 10 x) - (5 + 6 x - 5 x^{2}) \cdot 1}{x^{2}}$

Step 3.2.11

Multiply $- 1$ by $1$ .

$\frac{x (6 - 10 x) - (5 + 6 x - 5 x^{2})}{x^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{x \cdot 6 + x (- 10 x) - (5 + 6 x - 5 x^{2})}{x^{2}}$

Step 3.3.2

Apply the distributive property.

$\frac{x \cdot 6 + x (- 10 x) - 1 \cdot 5 - (6 x) - (- 5 x^{2})}{x^{2}}$

Step 3.3.3

Simplify the numerator.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move $6$ to the left of $x$ .

$\frac{6 \cdot x + x (- 10 x) - 1 \cdot 5 - (6 x) - (- 5 x^{2})}{x^{2}}$

Step 3.3.3.1.2

Rewrite using the commutative property of multiplication.

$\frac{6 x - 10 x \cdot x - 1 \cdot 5 - (6 x) - (- 5 x^{2})}{x^{2}}$

Step 3.3.3.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.3.1.3.1

Move $x$ .

$\frac{6 x - 10 (x \cdot x) - 1 \cdot 5 - (6 x) - (- 5 x^{2})}{x^{2}}$

Step 3.3.3.1.3.2

Multiply $x$ by $x$ .

$\frac{6 x - 10 x^{2} - 1 \cdot 5 - (6 x) - (- 5 x^{2})}{x^{2}}$

Step 3.3.3.1.4

Multiply $- 1$ by $5$ .

$\frac{6 x - 10 x^{2} - 5 - (6 x) - (- 5 x^{2})}{x^{2}}$

Step 3.3.3.1.5

Multiply $6$ by $- 1$ .

$\frac{6 x - 10 x^{2} - 5 - 6 x - (- 5 x^{2})}{x^{2}}$

Step 3.3.3.1.6

Multiply $- 5$ by $- 1$ .

$\frac{6 x - 10 x^{2} - 5 - 6 x + 5 x^{2}}{x^{2}}$

Step 3.3.3.2

Combine the opposite terms in $6 x - 10 x^{2} - 5 - 6 x + 5 x^{2}$ .

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Step 3.3.3.2.1

Subtract $6 x$ from $6 x$ .

$\frac{- 10 x^{2} - 5 + 0 + 5 x^{2}}{x^{2}}$

Step 3.3.3.2.2

Add $- 10 x^{2} - 5$ and $0$ .

$\frac{- 10 x^{2} - 5 + 5 x^{2}}{x^{2}}$

Step 3.3.3.3

Add $- 10 x^{2}$ and $5 x^{2}$ .

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

Factor $5$ out of $- 5$ .

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

Step 3.3.4.3

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

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

Step 3.3.5

Factor $- 1$ out of $- x^{2}$ .

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

Step 3.3.6

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

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

Step 3.3.7

Factor $- 1$ out of $- (x^{2}) - 1 (1)$ .

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

Step 3.3.8

Rewrite $- (x^{2} + 1)$ as $- 1 (x^{2} + 1)$ .

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

Step 3.3.9

Move the negative in front of the fraction.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{5 (x^{2} + 1)}{x^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{5 (x^{2} + 1)}{x^{2}}$