Calculus Examples

$\frac{x + 3}{x^{2}}$

Step 1

Write $\frac{x + 3}{x^{2}}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [x + 3] - (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.2

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

$\frac{x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.3

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

$\frac{x^{2} (1 + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4

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

$\frac{x^{2} (1 + 0) - (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.5

Simplify the expression.

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Step 2.1.2.5.1

Add $1$ and $0$ .

$\frac{x^{2} \cdot 1 - (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.5.2

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

$\frac{x^{2} - (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.6

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

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

Step 2.1.2.7

Simplify with factoring out.

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Step 2.1.2.7.1

Multiply $2$ by $- 1$ .

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

Step 2.1.2.7.2

Factor $x$ out of $x^{2} - 2 (x + 3) x$ .

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Step 2.1.2.7.2.1

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

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

Step 2.1.2.7.2.2

Factor $x$ out of $- 2 (x + 3) x$ .

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

Step 2.1.2.7.2.3

Factor $x$ out of $x \cdot x + x (- 2 (x + 3))$ .

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

Step 2.1.3

Cancel the common factors.

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Step 2.1.3.1

Factor $x$ out of $x^{4}$ .

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

Step 2.1.3.2

Cancel the common factor.

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

Step 2.1.3.3

Rewrite the expression.

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

Step 2.1.4

Simplify.

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Step 2.1.4.1

Apply the distributive property.

$\frac{x - 2 x - 2 \cdot 3}{x^{3}}$

Step 2.1.4.2

Simplify the numerator.

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Step 2.1.4.2.1

Multiply $- 2$ by $3$ .

$\frac{x - 2 x - 6}{x^{3}}$

Step 2.1.4.2.2

Subtract $2 x$ from $x$ .

$\frac{- x - 6}{x^{3}}$

Step 2.1.4.3

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

$\frac{- (x) - 6}{x^{3}}$

Step 2.1.4.4

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- (x) - 1 (6)}{x^{3}}$

Step 2.1.4.5

Factor $- 1$ out of $- (x) - 1 (6)$ .

$\frac{- (x + 6)}{x^{3}}$

Step 2.1.4.6

Rewrite $- (x + 6)$ as $- 1 (x + 6)$ .

$\frac{- 1 (x + 6)}{x^{3}}$

Step 2.1.4.7

Move the negative in front of the fraction.

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

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{x + 6}{x^{3}}$ .

$- \frac{d}{d x} [\frac{x + 6}{x^{3}}] + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.2

$- \frac{x^{3} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3

Differentiate.

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Step 2.2.3.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.2.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{x^{3} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.1.2

Multiply $3$ by $2$ .

$- \frac{x^{3} \frac{d}{d x} [x + 6] - (x + 6) \frac{d}{d x} [x^{3}]}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.2

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

$- \frac{x^{3} (\frac{d}{d x} [x] + \frac{d}{d x} [6]) - (x + 6) \frac{d}{d x} [x^{3}]}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.3

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

$- \frac{x^{3} (1 + \frac{d}{d x} [6]) - (x + 6) \frac{d}{d x} [x^{3}]}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$- \frac{x^{3} (1 + 0) - (x + 6) \frac{d}{d x} [x^{3}]}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.5

Simplify the expression.

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Step 2.2.3.5.1

Add $1$ and $0$ .

$- \frac{x^{3} \cdot 1 - (x + 6) \frac{d}{d x} [x^{3}]}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.5.2

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

$- \frac{x^{3} - (x + 6) \frac{d}{d x} [x^{3}]}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- \frac{x^{3} - (x + 6) (3 x^{2})}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.7

Simplify with factoring out.

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Step 2.2.3.7.1

Multiply $3$ by $- 1$ .

$- \frac{x^{3} - 3 (x + 6) x^{2}}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.7.2

Factor $x^{2}$ out of $x^{3} - 3 (x + 6) x^{2}$ .

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Step 2.2.3.7.2.1

Factor $x^{2}$ out of $x^{3}$ .

$- \frac{x^{2} x - 3 (x + 6) x^{2}}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.7.2.2

Factor $x^{2}$ out of $- 3 (x + 6) x^{2}$ .

$- \frac{x^{2} x + x^{2} (- 3 (x + 6))}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.3.7.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} (- 3 (x + 6))$ .

$- \frac{x^{2} (x - 3 (x + 6))}{x^{6}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.4

Cancel the common factors.

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Step 2.2.4.1

Factor $x^{2}$ out of $x^{6}$ .

$- \frac{x^{2} (x - 3 (x + 6))}{x^{2} x^{4}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.4.2

Cancel the common factor.

$- \frac{x^{2} (x - 3 (x + 6))}{x^{2} x^{4}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.4.3

Rewrite the expression.

$- \frac{x - 3 (x + 6)}{x^{4}} + \frac{x + 6}{x^{3}} \frac{d}{d x} [- 1]$

Step 2.2.5

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

$- \frac{x - 3 (x + 6)}{x^{4}} + \frac{x + 6}{x^{3}} \cdot 0$

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Multiply $\frac{x + 6}{x^{3}}$ by $0$ .

$- \frac{x - 3 (x + 6)}{x^{4}} + 0$

Step 2.2.6.2

Add $- \frac{x - 3 (x + 6)}{x^{4}}$ and $0$ .

$- \frac{x - 3 (x + 6)}{x^{4}}$

Step 2.2.7

Simplify.

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Step 2.2.7.1

Apply the distributive property.

$- \frac{x - 3 x - 3 \cdot 6}{x^{4}}$

Step 2.2.7.2

Simplify the numerator.

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Step 2.2.7.2.1

Multiply $- 3$ by $6$ .

$- \frac{x - 3 x - 18}{x^{4}}$

Step 2.2.7.2.2

Subtract $3 x$ from $x$ .

$- \frac{- 2 x - 18}{x^{4}}$

Step 2.2.7.3

Factor $2$ out of $- 2 x - 18$ .

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Step 2.2.7.3.1

Factor $2$ out of $- 2 x$ .

$- \frac{2 (- x) - 18}{x^{4}}$

Step 2.2.7.3.2

Factor $2$ out of $- 18$ .

$- \frac{2 (- x) + 2 (- 9)}{x^{4}}$

Step 2.2.7.3.3

Factor $2$ out of $2 (- x) + 2 (- 9)$ .

$- \frac{2 (- x - 9)}{x^{4}}$

Step 2.2.7.4

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

$- \frac{2 (- (x) - 9)}{x^{4}}$

Step 2.2.7.5

Rewrite $- 9$ as $- 1 (9)$ .

$- \frac{2 (- (x) - 1 (9))}{x^{4}}$

Step 2.2.7.6

Factor $- 1$ out of $- (x) - 1 (9)$ .

$- \frac{2 (- (x + 9))}{x^{4}}$

Step 2.2.7.7

Rewrite $- (x + 9)$ as $- 1 (x + 9)$ .

$- \frac{2 (- 1 (x + 9))}{x^{4}}$

Step 2.2.7.8

Move the negative in front of the fraction.

$- - \frac{2 (x + 9)}{x^{4}}$

Step 2.2.7.9

Multiply $- 1$ by $- 1$ .

$1 \frac{2 (x + 9)}{x^{4}}$

Step 2.2.7.10

Multiply $\frac{2 (x + 9)}{x^{4}}$ by $1$ .

$f'' (x) = \frac{2 (x + 9)}{x^{4}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 (x + 9)}{x^{4}}$ .

$\frac{2 (x + 9)}{x^{4}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{2 (x + 9)}{x^{4}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{2 (x + 9)}{x^{4}} = 0$

Step 3.2

Set the numerator equal to zero.

$2 (x + 9) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Divide each term in $2 (x + 9) = 0$ by $2$ and simplify.

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Step 3.3.1.1

Divide each term in $2 (x + 9) = 0$ by $2$ .

$\frac{2 (x + 9)}{2} = \frac{0}{2}$

Step 3.3.1.2

Simplify the left side.

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Step 3.3.1.2.1

Cancel the common factor of $2$ .

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Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{2 (x + 9)}{2} = \frac{0}{2}$

Step 3.3.1.2.1.2

Divide $x + 9$ by $1$ .

$x + 9 = \frac{0}{2}$

Step 3.3.1.3

Simplify the right side.

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Step 3.3.1.3.1

Divide $0$ by $2$ .

$x + 9 = 0$

Step 3.3.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $- 9$ in $f (x) = \frac{x + 3}{x^{2}}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $- 9$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Add $- 9$ and $3$ .

$f (- 9) = \frac{- 6}{{(- 9)}^{2}}$

Step 4.1.2.1.2

Raise $- 9$ to the power of $2$ .

$f (- 9) = \frac{- 6}{81}$

Step 4.1.2.2

Cancel the common factor of $- 6$ and $81$ .

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Step 4.1.2.2.1

Factor $3$ out of $- 6$ .

$f (- 9) = \frac{3 (- 2)}{81}$

Step 4.1.2.2.2

Cancel the common factors.

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Step 4.1.2.2.2.1

Factor $3$ out of $81$ .

$f (- 9) = \frac{3 \cdot - 2}{3 \cdot 27}$

Step 4.1.2.2.2.2

Cancel the common factor.

$f (- 9) = \frac{3 \cdot - 2}{3 \cdot 27}$

Step 4.1.2.2.2.3

Rewrite the expression.

$f (- 9) = \frac{- 2}{27}$

Step 4.1.2.3

Move the negative in front of the fraction.

$f (- 9) = - \frac{2}{27}$

Step 4.1.2.4

The final answer is $- \frac{2}{27}$ .

$- \frac{2}{27}$

Step 4.2

The point found by substituting $- 9$ in $f (x) = \frac{x + 3}{x^{2}}$ is $(- 9, - \frac{2}{27})$ . This point can be an inflection point.

$(- 9, - \frac{2}{27})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 9) \cup (- 9, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 9)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 9.1$ in the expression.

$f'' (- 9.1) = \frac{2 ((- 9.1) + 9)}{{(- 9.1)}^{4}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Add $- 9.1$ and $9$ .

$f'' (- 9.1) = \frac{2 \cdot - 0.1}{{(- 9.1)}^{4}}$

Step 6.2.2

Raise $- 9.1$ to the power of $4$ .

$f'' (- 9.1) = \frac{2 \cdot - 0.1}{6857.4961}$

Step 6.2.3

Multiply $2$ by $- 0.1$ .

$f'' (- 9.1) = \frac{- 0.2}{6857.4961}$

Step 6.2.4

Divide $- 0.2$ by $6857.4961$ .

$f'' (- 9.1) = - 0.00002916$

Step 6.2.5

The final answer is $- 0.00002916$ .

$- 0.00002916$

Step 6.3

At $- 9.1$ , the second derivative is $- 2.91651642 E - 5$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 9)$

Decreasing on $(- \infty, - 9)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 9, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 8.9$ in the expression.

$f'' (- 8.9) = \frac{2 ((- 8.9) + 9)}{{(- 8.9)}^{4}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Add $- 8.9$ and $9$ .

$f'' (- 8.9) = \frac{2 \cdot 0.1}{{(- 8.9)}^{4}}$

Step 7.2.2

Raise $- 8.9$ to the power of $4$ .

$f'' (- 8.9) = \frac{2 \cdot 0.1}{6274.2241}$

Step 7.2.3

Multiply $2$ by $0.1$ .

$f'' (- 8.9) = \frac{0.2}{6274.2241}$

Step 7.2.4

Divide $0.2$ by $6274.2241$ .

$f'' (- 8.9) = 0.00003187$

Step 7.2.5

The final answer is $0.00003187$ .

$0.00003187$

Step 7.3

At $- 8.9$ , the second derivative is $3.18764514 E - 5$ . Since this is positive, the second derivative is increasing on the interval $(- 9, \infty)$ .

Increasing on $(- 9, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(- 9, - \frac{2}{27})$ .

$(- 9, - \frac{2}{27})$

Step 9