Calculus Examples

$\frac{3 x - 4}{x^{2} + 1}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $f (x) = 3 x - 4$ and $g (x) = x^{2} + 1$ .

$f' (x) = \frac{(x^{2} + 1) \frac{d}{d x} (3 x - 4) - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 x - 4$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$ .

$f' (x) = \frac{(x^{2} + 1) (\frac{d}{d x} (3 x) + \frac{d}{d x} (- 4)) - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.2

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

$f' (x) = \frac{(x^{2} + 1) (3 \frac{d}{d x} (x) + \frac{d}{d x} (- 4)) - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.3

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

$f' (x) = \frac{(x^{2} + 1) (3 \cdot 1 + \frac{d}{d x} (- 4)) - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.4

Multiply $3$ by $1$ .

$f' (x) = \frac{(x^{2} + 1) (3 + \frac{d}{d x} (- 4)) - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f' (x) = \frac{(x^{2} + 1) (3 + 0) - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $3$ and $0$ .

$f' (x) = \frac{(x^{2} + 1) \cdot 3 - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.6.2

Move $3$ to the left of $x^{2} + 1$ .

$f' (x) = \frac{3 \cdot (x^{2} + 1) - (3 x - 4) \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.7

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

$f' (x) = \frac{3 (x^{2} + 1) - (3 x - 4) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.8

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

$f' (x) = \frac{3 (x^{2} + 1) - (3 x - 4) (2 x + \frac{d}{d x} (1))}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.9

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

$f' (x) = \frac{3 (x^{2} + 1) - (3 x - 4) (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.10

Simplify the expression.

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Step 1.1.2.10.1

Add $2 x$ and $0$ .

$f' (x) = \frac{3 (x^{2} + 1) - (3 x - 4) (2 x)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.10.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{3 (x^{2} + 1) - 2 (3 x - 4) x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$f' (x) = \frac{3 x^{2} + 3 \cdot 1 - 2 (3 x - 4) x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$f' (x) = \frac{3 x^{2} + 3 \cdot 1 + (- 2 (3 x) - 2 \cdot - 4) x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.3

Apply the distributive property.

$f' (x) = \frac{3 x^{2} + 3 \cdot 1 - 2 (3 x) x - 2 \cdot (- 4 x)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.4

Simplify the numerator.

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Step 1.1.3.4.1

Simplify each term.

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Step 1.1.3.4.1.1

Multiply $3$ by $1$ .

$f' (x) = \frac{3 x^{2} + 3 - 2 (3 x) x - 2 \cdot (- 4 x)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.4.1.2

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

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Step 1.1.3.4.1.2.1

Move $x$ .

$f' (x) = \frac{3 x^{2} + 3 - 2 (3 (x \cdot x)) - 2 \cdot (- 4 x)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.4.1.2.2

Multiply $x$ by $x$ .

$f' (x) = \frac{3 x^{2} + 3 - 2 (3 x^{2}) - 2 \cdot (- 4 x)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.4.1.3

Multiply $3$ by $- 2$ .

$f' (x) = \frac{3 x^{2} + 3 - 6 x^{2} - 2 \cdot (- 4 x)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.4.1.4

Multiply $- 2$ by $- 4$ .

$f' (x) = \frac{3 x^{2} + 3 - 6 x^{2} + 8 x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.4.2

Subtract $6 x^{2}$ from $3 x^{2}$ .

$f' (x) = \frac{- 3 x^{2} + 3 + 8 x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.5

Reorder terms.

$f' (x) = \frac{- 3 x^{2} + 8 x + 3}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.6

Factor by grouping.

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Step 1.1.3.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot 3 = - 9$ and whose sum is $b = 8$ .

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Step 1.1.3.6.1.1

Factor $8$ out of $8 x$ .

$f' (x) = \frac{- 3 x^{2} + 8 (x) + 3}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.6.1.2

Rewrite $8$ as $- 1$ plus $9$

$f' (x) = \frac{- 3 x^{2} + (- 1 + 9) x + 3}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.6.1.3

Apply the distributive property.

$f' (x) = \frac{- 3 x^{2} - 1 x + 9 x + 3}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.6.2

Factor out the greatest common factor from each group.

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Step 1.1.3.6.2.1

Group the first two terms and the last two terms.

$f' (x) = \frac{(- 3 x^{2} - 1 x) + 9 x + 3}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.6.2.2

Factor out the greatest common factor (GCF) from each group.

$f' (x) = \frac{x (- 3 x - 1) - 3 (- 3 x - 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.6.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x - 1$ .

$f' (x) = \frac{(- 3 x - 1) (x - 3)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.7

Factor $- 1$ out of $- 3 x$ .

$f' (x) = \frac{(- (3 x) - 1) (x - 3)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.8

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

$f' (x) = \frac{(- (3 x) - 1 \cdot 1) (x - 3)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.9

Factor $- 1$ out of $- (3 x) - 1 (1)$ .

$f' (x) = \frac{- (3 x + 1) (x - 3)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.10

Rewrite $- (3 x + 1)$ as $- 1 (3 x + 1)$ .

$f' (x) = \frac{- 1 (3 x + 1) (x - 3)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.11

Move the negative in front of the fraction.

$f' (x) = - \frac{(3 x + 1) (x - 3)}{{(x^{2} + 1)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{(3 x + 1) (x - 3)}{{(x^{2} + 1)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{(3 x + 1) (x - 3)}{{(x^{2} + 1)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{(3 x + 1) (x - 3)}{{(x^{2} + 1)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$(3 x + 1) (x - 3) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x + 1 = 0$

$x - 3 = 0$

Step 2.3.2

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 2.3.2.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 2.3.2.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 2.3.2.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 2.3.2.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.3.2.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 2.3.2.2.2.3

Simplify the right side.

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Step 2.3.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 2.3.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.3.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.3.4

The final solution is all the values that make $(3 x + 1) (x - 3) = 0$ true.

$x = - \frac{1}{3}, 3$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{3 x - 4}{x^{2} + 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{1}{3}$ .

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Step 4.1.1

Substitute $- \frac{1}{3}$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 4.1.2.1.1.2

Cancel the common factor.

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

Step 4.1.2.1.1.3

Rewrite the expression.

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

Step 4.1.2.1.2

Subtract $4$ from $- 1$ .

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

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.2.1.1

Apply the product rule to $- \frac{1}{3}$ .

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

Step 4.1.2.2.1.2

Apply the product rule to $\frac{1}{3}$ .

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

Step 4.1.2.2.2

Raise $- 1$ to the power of $2$ .

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

Step 4.1.2.2.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

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

Step 4.1.2.2.4

One to any power is one.

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

Step 4.1.2.2.5

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

$\frac{- 5}{\frac{1}{9} + 1}$

Step 4.1.2.2.6

Write $1$ as a fraction with a common denominator.

$\frac{- 5}{\frac{1}{9} + \frac{9}{9}}$

Step 4.1.2.2.7

Combine the numerators over the common denominator.

$\frac{- 5}{\frac{1 + 9}{9}}$

Step 4.1.2.2.8

Add $1$ and $9$ .

$\frac{- 5}{\frac{10}{9}}$

Step 4.1.2.3

Multiply the numerator by the reciprocal of the denominator.

$- 5 (\frac{9}{10})$

Step 4.1.2.4

Cancel the common factor of $5$ .

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Step 4.1.2.4.1

Factor $5$ out of $- 5$ .

$5 (- 1) \frac{9}{10}$

Step 4.1.2.4.2

Factor $5$ out of $10$ .

$5 \cdot - 1 \frac{9}{5 \cdot 2}$

Step 4.1.2.4.3

Cancel the common factor.

$5 \cdot - 1 \frac{9}{5 \cdot 2}$

Step 4.1.2.4.4

Rewrite the expression.

$- \frac{9}{2}$

Step 4.2

Evaluate at $x = 3$ .

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Step 4.2.1

Substitute $3$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Multiply $3$ by $3$ .

$\frac{9 - 4}{3^{2} + 1}$

Step 4.2.2.1.2

Subtract $4$ from $9$ .

$\frac{5}{3^{2} + 1}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

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

$\frac{5}{9 + 1}$

Step 4.2.2.2.2

Add $9$ and $1$ .

$\frac{5}{10}$

Step 4.2.2.3

Cancel the common factor of $5$ and $10$ .

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Step 4.2.2.3.1

Factor $5$ out of $5$ .

$\frac{5 (1)}{10}$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 1}{5 \cdot 2}$

Step 4.2.2.3.2.2

Cancel the common factor.

$\frac{5 \cdot 1}{5 \cdot 2}$

Step 4.2.2.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 4.3

List all of the points.

$(- \frac{1}{3}, - \frac{9}{2}), (3, \frac{1}{2})$

Step 5