Calculus Examples

$\frac{4 x - 5}{x + 1}$ $x = 0$

Step 1

Write $\frac{4 x - 5}{x + 1}$ as an equation.

$y = \frac{4 x - 5}{x + 1}$

Step 2

Find the corresponding $y$ -value to $x = 0$ .

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Step 2.1

Substitute $0$ in for $x$ .

$y = \frac{4 (0) - 5}{(0) + 1}$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Remove parentheses.

$y = \frac{4 (0) - 5}{0 + 1}$

Step 2.2.2

Remove parentheses.

$y = \frac{4 (0) - 5}{(0) + 1}$

Step 2.2.3

Simplify $\frac{4 (0) - 5}{(0) + 1}$ .

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Step 2.2.3.1

Simplify the numerator.

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Step 2.2.3.1.1

Multiply $4$ by $0$ .

$y = \frac{0 - 5}{0 + 1}$

Step 2.2.3.1.2

Subtract $5$ from $0$ .

$y = \frac{- 5}{0 + 1}$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Add $0$ and $1$ .

$y = \frac{- 5}{1}$

Step 2.2.3.2.2

Divide $- 5$ by $1$ .

$y = - 5$

Step 3

Find the first derivative and evaluate at $x = 0$ and $y = - 5$ to find the slope of the tangent line.

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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 $f (x) = 4 x - 5$ and $g (x) = x + 1$ .

$\frac{(x + 1) \frac{d}{d x} [4 x - 5] - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $4 x - 5$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5]$ .

$\frac{(x + 1) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5]) - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{(x + 1) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]) - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2.3

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

$\frac{(x + 1) (4 \cdot 1 + \frac{d}{d x} [- 5]) - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2.4

Multiply $4$ by $1$ .

$\frac{(x + 1) (4 + \frac{d}{d x} [- 5]) - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2.5

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

$\frac{(x + 1) (4 + 0) - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $4$ and $0$ .

$\frac{(x + 1) \cdot 4 - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2.6.2

Move $4$ to the left of $x + 1$ .

$\frac{4 \cdot (x + 1) - (4 x - 5) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3.2.7

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

$\frac{4 (x + 1) - (4 x - 5) (\frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 3.2.8

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

$\frac{4 (x + 1) - (4 x - 5) (1 + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 3.2.9

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

$\frac{4 (x + 1) - (4 x - 5) (1 + 0)}{{(x + 1)}^{2}}$

Step 3.2.10

Simplify the expression.

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Step 3.2.10.1

Add $1$ and $0$ .

$\frac{4 (x + 1) - (4 x - 5) \cdot 1}{{(x + 1)}^{2}}$

Step 3.2.10.2

Multiply $- 1$ by $1$ .

$\frac{4 (x + 1) - (4 x - 5)}{{(x + 1)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{4 x + 4 \cdot 1 - (4 x - 5)}{{(x + 1)}^{2}}$

Step 3.3.2

Apply the distributive property.

$\frac{4 x + 4 \cdot 1 - (4 x) - - 5}{{(x + 1)}^{2}}$

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

Multiply $4$ by $1$ .

$\frac{4 x + 4 - (4 x) - - 5}{{(x + 1)}^{2}}$

Step 3.3.3.1.2

Multiply $4$ by $- 1$ .

$\frac{4 x + 4 - 4 x - - 5}{{(x + 1)}^{2}}$

Step 3.3.3.1.3

Multiply $- 1$ by $- 5$ .

$\frac{4 x + 4 - 4 x + 5}{{(x + 1)}^{2}}$

Step 3.3.3.2

Combine the opposite terms in $4 x + 4 - 4 x + 5$ .

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Step 3.3.3.2.1

Subtract $4 x$ from $4 x$ .

$\frac{0 + 4 + 5}{{(x + 1)}^{2}}$

Step 3.3.3.2.2

Add $0$ and $4$ .

$\frac{4 + 5}{{(x + 1)}^{2}}$

Step 3.3.3.3

Add $4$ and $5$ .

$\frac{9}{{(x + 1)}^{2}}$

Step 3.4

Evaluate the derivative at $x = 0$ .

$\frac{9}{{((0) + 1)}^{2}}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the denominator.

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Step 3.5.1.1

Add $0$ and $1$ .

$\frac{9}{1^{2}}$

Step 3.5.1.2

One to any power is one.

$\frac{9}{1}$

Step 3.5.2

Divide $9$ by $1$ .

$9$

Step 4

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 4.1

Use the slope $9$ and a given point $(0, - 5)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 5) = 9 \cdot (x - (0))$

Step 4.2

Simplify the equation and keep it in point-slope form.

$y + 5 = 9 \cdot (x + 0)$

Step 4.3

Solve for $y$ .

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Step 4.3.1

Add $x$ and $0$ .

$y + 5 = 9 x$

Step 4.3.2

Subtract $5$ from both sides of the equation.

$y = 9 x - 5$

Step 5