Calculus Examples

$f (x) = \frac{1}{x + 1}$ , $(0, 1)$

Step 1

Check if the given point is on the graph of the given function.

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Step 1.1

Evaluate $f (x) = \frac{1}{x + 1}$ at $x = 0$ .

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Step 1.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{1}{(0) + 1}$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Add $0$ and $1$ .

$f (0) = \frac{1}{1}$

Step 1.1.2.2

Divide $1$ by $1$ .

$f (0) = 1$

Step 1.1.2.3

The final answer is $1$ .

$1$

Step 1.2

Since $1 = 1$ , the point is on the graph.

The point is on the graph

Step 2

The slope of the tangent line is the derivative of the expression.

$m$ $=$ The derivative of $f (x) = \frac{1}{x + 1}$

Step 3

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 4

Find the components of the definition.

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Step 4.1

Evaluate the function at $x = x + h$ .

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Step 4.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{1}{(x + h) + 1}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Remove parentheses.

$f (x + h) = \frac{1}{x + h + 1}$

Step 4.1.2.2

The final answer is $\frac{1}{x + h + 1}$ .

$\frac{1}{x + h + 1}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{1}{x + h + 1}$

$f (x) = \frac{1}{x + 1}$

$f (x + h) = \frac{1}{x + h + 1}$

$f (x) = \frac{1}{x + 1}$

Step 5

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\frac{1}{x + h + 1} - (\frac{1}{x + 1})}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

To write $\frac{1}{x + h + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{1}{x + h + 1} \cdot \frac{x + 1}{x + 1} - \frac{1}{x + 1}}{h}$

Step 6.1.2

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

$f' (x) = lim_{h \to 0} \frac{\frac{1}{x + h + 1} \cdot \frac{x + 1}{x + 1} - \frac{1}{x + 1} \cdot \frac{x + h + 1}{x + h + 1}}{h}$

Step 6.1.3

Write each expression with a common denominator of $(x + h + 1) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1.3.1

Multiply $\frac{1}{x + h + 1}$ by $\frac{x + 1}{x + 1}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x + 1}{(x + h + 1) (x + 1)} - \frac{1}{x + 1} \cdot \frac{x + h + 1}{x + h + 1}}{h}$

Step 6.1.3.2

Multiply $\frac{1}{x + 1}$ by $\frac{x + h + 1}{x + h + 1}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x + 1}{(x + h + 1) (x + 1)} - \frac{x + h + 1}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.3.3

Reorder the factors of $(x + h + 1) (x + 1)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x + 1}{(x + 1) (x + h + 1)} - \frac{x + h + 1}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.4

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{x + 1 - (x + h + 1)}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.5

Rewrite $\frac{x + 1 - (x + h + 1)}{(x + 1) (x + h + 1)}$ in a factored form.

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Step 6.1.5.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{x + 1 - x - h - 1 \cdot 1}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.5.2

Multiply $- 1$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x + 1 - x - h - 1}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.5.3

Subtract $x$ from $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{0 + 1 - h - 1}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.5.4

Add $0$ and $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{1 - h - 1}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.5.5

Subtract $1$ from $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{- h + 0}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.5.6

Add $- h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{- h}{(x + 1) (x + h + 1)}}{h}$

Step 6.1.6

Move the negative in front of the fraction.

$f' (x) = lim_{h \to 0} \frac{- \frac{h}{(x + 1) (x + h + 1)}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$f' (x) = lim_{h \to 0} - \frac{h}{(x + 1) (x + h + 1)} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Move the leading negative in $- \frac{h}{(x + 1) (x + h + 1)}$ into the numerator.

$f' (x) = lim_{h \to 0} \frac{- h}{(x + 1) (x + h + 1)} \cdot \frac{1}{h}$

Step 6.3.2

Factor $h$ out of $- h$ .

$f' (x) = lim_{h \to 0} \frac{h \cdot - 1}{(x + 1) (x + h + 1)} \cdot \frac{1}{h}$

Step 6.3.3

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h \cdot - 1}{(x + 1) (x + h + 1)} \cdot \frac{1}{h}$

Step 6.3.4

Rewrite the expression.

$f' (x) = lim_{h \to 0} \frac{- 1}{(x + 1) (x + h + 1)}$

Step 6.4

Move the negative in front of the fraction.

$f' (x) = lim_{h \to 0} - \frac{1}{(x + 1) (x + h + 1)}$

Step 7

Evaluate the limit.

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Step 7.1

Move the term $- 1$ outside of the limit because it is constant with respect to $h$ .

$- lim_{h \to 0} \frac{1}{(x + 1) (x + h + 1)}$

Step 7.2

Move the term $\frac{1}{x + 1}$ outside of the limit because it is constant with respect to $h$ .

$- \frac{1}{x + 1} lim_{h \to 0} \frac{1}{x + h + 1}$

Step 7.3

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$- \frac{1}{x + 1} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} x + h + 1}$

Step 7.4

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$- \frac{1}{x + 1} \cdot \frac{1}{lim_{h \to 0} x + h + 1}$

Step 7.5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$- \frac{1}{x + 1} \cdot \frac{1}{lim_{h \to 0} x + lim_{h \to 0} h + lim_{h \to 0} 1}$

Step 7.6

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$- \frac{1}{x + 1} \cdot \frac{1}{x + lim_{h \to 0} h + lim_{h \to 0} 1}$

Step 7.7

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$- \frac{1}{x + 1} \cdot \frac{1}{x + lim_{h \to 0} h + 1}$

Step 8

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$- \frac{1}{x + 1} \cdot \frac{1}{x + 0 + 1}$

Step 9

Simplify the answer.

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Step 9.1

Add $x$ and $0$ .

$- \frac{1}{x + 1} \cdot \frac{1}{x + 1}$

Step 9.2

Multiply $- \frac{1}{x + 1} \cdot \frac{1}{x + 1}$ .

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Step 9.2.1

Multiply $\frac{1}{x + 1}$ by $\frac{1}{x + 1}$ .

$- \frac{1}{(x + 1) (x + 1)}$

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

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

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

Step 9.2.5

Add $1$ and $1$ .

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

Step 10

Find the slope $m$ . In this case $m = - 1$ .

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Step 10.1

Remove parentheses.

$m = - \frac{1}{{(0 + 1)}^{2}}$

Step 10.2

Remove parentheses.

$m = - \frac{1}{{((0) + 1)}^{2}}$

Step 10.3

Simplify $- \frac{1}{{((0) + 1)}^{2}}$ .

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Step 10.3.1

Simplify the denominator.

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Step 10.3.1.1

Add $0$ and $1$ .

$m = - \frac{1}{1^{2}}$

Step 10.3.1.2

One to any power is one.

$m = - \frac{1}{1}$

Step 10.3.2

Reduce the expression by cancelling the common factors.

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Step 10.3.2.1

Cancel the common factor of $1$ .

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Step 10.3.2.1.1

Cancel the common factor.

$m = - \frac{1}{1}$

Step 10.3.2.1.2

Rewrite the expression.

$m = - 1 \cdot 1$

Step 10.3.2.2

Multiply $- 1$ by $1$ .

$m = - 1$

Step 11

The slope is $m = - 1$ and the point is $(0, 1)$ .

$m = - 1, (0, 1)$

Step 12

Find the value of $b$ using the formula for the equation of a line.

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Step 12.1

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Step 12.2

Substitute the value of $m$ into the equation.

$y = (- 1) \cdot x + b$

Step 12.3

Substitute the value of $x$ into the equation.

$y = (- 1) \cdot (0) + b$

Step 12.4

Substitute the value of $y$ into the equation.

$1 = (- 1) \cdot (0) + b$

Step 12.5

Find the value of $b$ .

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Step 12.5.1

Rewrite the equation as $(- 1) \cdot (0) + b = 1$ .

$(- 1) \cdot (0) + b = 1$

Step 12.5.2

Simplify $(- 1) \cdot (0) + b$ .

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Step 12.5.2.1

Multiply $- 1$ by $0$ .

$0 + b = 1$

Step 12.5.2.2

Add $0$ and $b$ .

$b = 1$

Step 13

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = - x + 1$

Step 14