Calculus Examples

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

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

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

$\frac{(x^{2} + 3) (1 + 0) - (x - 1) \frac{d}{d x} [x^{2} + 3]}{{(x^{2} + 3)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

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

Step 1.1.2.4.2

Multiply $x^{2} + 3$ by $1$ .

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

Step 1.1.2.5

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

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

Step 1.1.2.6

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

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

Step 1.1.2.7

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $2 x$ and $0$ .

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

Step 1.1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Apply the distributive property.

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

Step 1.1.3.3

Simplify the numerator.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

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Step 1.1.3.3.1.1.1

Move $x$ .

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

Step 1.1.3.3.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.3.3.1.2

Multiply $- 2$ by $- 1$ .

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

Step 1.1.3.3.2

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

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

Step 1.1.3.4

Reorder terms.

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

Step 1.1.3.5

Factor by grouping.

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Step 1.1.3.5.1

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 $a \cdot c = - 1 \cdot 3 = - 3$ and whose sum is $b = 2$ .

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Step 1.1.3.5.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.3.5.1.2

Rewrite $2$ as $- 1$ plus $3$

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

Step 1.1.3.5.1.3

Apply the distributive property.

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

Step 1.1.3.5.2

Factor out the greatest common factor from each group.

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Step 1.1.3.5.2.1

Group the first two terms and the last two terms.

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

Step 1.1.3.5.2.2

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

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

Step 1.1.3.5.3

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

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

Step 1.1.3.6

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

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

Step 1.1.3.7

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

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

Step 1.1.3.8

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

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

Step 1.1.3.9

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

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

Step 1.1.3.10

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

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

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 $0$ .

$x + 1 = 0$

$x - 3 = 0$

Step 2.3.2

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

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Step 2.3.2.1

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

$x + 1 = 0$

Step 2.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$x = - 1, 3$

Step 3

The values which make the derivative equal to $0$ are $- 1, 3$ .

$- 1, 3$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 1) \cup (- 1, 3) \cup (3, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Add $- 2$ and $1$ .

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

Step 5.2.1.2

Subtract $3$ from $- 2$ .

$f' (- 2) = - \frac{- 1 \cdot - 5}{{({(- 2)}^{2} + 3)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

$f' (- 2) = - \frac{- 1 \cdot - 5}{{(4 + 3)}^{2}}$

Step 5.2.2.2

Add $4$ and $3$ .

$f' (- 2) = - \frac{- 1 \cdot - 5}{7^{2}}$

Step 5.2.2.3

Raise $7$ to the power of $2$ .

$f' (- 2) = - \frac{- 1 \cdot - 5}{49}$

Step 5.2.3

Multiply $- 1$ by $- 5$ .

$f' (- 2) = - \frac{5}{49}$

Step 5.2.4

The final answer is $- \frac{5}{49}$ .

$- \frac{5}{49}$

Step 5.3

At $x = - 2$ the derivative is $- \frac{5}{49}$ . Since this is negative, the function is decreasing on $(- \infty, - 1)$ .

Decreasing on $(- \infty, - 1)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 1, 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Add $1$ and $1$ .

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

Step 6.2.1.2

Subtract $3$ from $1$ .

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

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

One to any power is one.

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

Step 6.2.2.2

Add $1$ and $3$ .

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

Step 6.2.2.3

Raise $4$ to the power of $2$ .

$f' (1) = - \frac{2 \cdot - 2}{16}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Multiply $2$ by $- 2$ .

$f' (1) = - \frac{- 4}{16}$

Step 6.2.3.2

Cancel the common factor of $- 4$ and $16$ .

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Step 6.2.3.2.1

Factor $4$ out of $- 4$ .

$f' (1) = - \frac{4 (- 1)}{16}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

Factor $4$ out of $16$ .

$f' (1) = - \frac{4 \cdot - 1}{4 \cdot 4}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f' (1) = - \frac{4 \cdot - 1}{4 \cdot 4}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f' (1) = - \frac{- 1}{4}$

Step 6.2.3.3

Move the negative in front of the fraction.

$f' (1) = \frac{1}{4}$

Step 6.2.4

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Step 6.3

At $x = 1$ the derivative is $\frac{1}{4}$ . Since this is positive, the function is increasing on $(- 1, 3)$ .

Increasing on $(- 1, 3)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $4$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Add $4$ and $1$ .

$f' (4) = - \frac{5 (4 - 3)}{{(4^{2} + 3)}^{2}}$

Step 7.2.1.2

Subtract $3$ from $4$ .

$f' (4) = - \frac{5 \cdot 1}{{(4^{2} + 3)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $4$ to the power of $2$ .

$f' (4) = - \frac{5 \cdot 1}{{(16 + 3)}^{2}}$

Step 7.2.2.2

Add $16$ and $3$ .

$f' (4) = - \frac{5 \cdot 1}{19^{2}}$

Step 7.2.2.3

Raise $19$ to the power of $2$ .

$f' (4) = - \frac{5 \cdot 1}{361}$

Step 7.2.3

Multiply $5$ by $1$ .

$f' (4) = - \frac{5}{361}$

Step 7.2.4

The final answer is $- \frac{5}{361}$ .

$- \frac{5}{361}$

Step 7.3

At $x = 4$ the derivative is $- \frac{5}{361}$ . Since this is negative, the function is decreasing on $(3, \infty)$ .

Decreasing on $(3, \infty)$ since $f' (x) < 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- 1, 3)$

Decreasing on: $(- \infty, - 1), (3, \infty)$

Step 9