Calculus Examples

$\frac{7 x^{2}}{x^{2} + 36}$

Step 1

Write $\frac{7 x^{2}}{x^{2} + 36}$ as a function.

$f (x) = \frac{7 x^{2}}{x^{2} + 36}$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Since $7$ is constant with respect to $x$ , the derivative of $\frac{7 x^{2}}{x^{2} + 36}$ with respect to $x$ is $7 \frac{d}{d x} [\frac{x^{2}}{x^{2} + 36}]$ .

$7 \frac{d}{d x} [\frac{x^{2}}{x^{2} + 36}]$

Step 2.1.2

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 $f (x) = x^{2}$ and $g (x) = x^{2} + 36$ .

$7 \frac{(x^{2} + 36) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

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

$7 \frac{(x^{2} + 36) (2 x) - x^{2} \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.1.3.2

Move $2$ to the left of $x^{2} + 36$ .

$7 \frac{2 \cdot (x^{2} + 36) x - x^{2} \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.1.3.3

By the Sum Rule, the derivative of $x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]$ .

$7 \frac{2 (x^{2} + 36) x - x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 2.1.3.4

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

$7 \frac{2 (x^{2} + 36) x - x^{2} (2 x + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 2.1.3.5

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$7 \frac{2 (x^{2} + 36) x - x^{2} (2 x + 0)}{{(x^{2} + 36)}^{2}}$

Step 2.1.3.6

Simplify the expression.

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Step 2.1.3.6.1

Add $2 x$ and $0$ .

$7 \frac{2 (x^{2} + 36) x - x^{2} (2 x)}{{(x^{2} + 36)}^{2}}$

Step 2.1.3.6.2

Multiply $2$ by $- 1$ .

$7 \frac{2 (x^{2} + 36) x - 2 x^{2} x}{{(x^{2} + 36)}^{2}}$

Step 2.1.4

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

$7 \frac{2 (x^{2} + 36) x - 2 (x^{1} x^{2})}{{(x^{2} + 36)}^{2}}$

Step 2.1.5

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

$7 \frac{2 (x^{2} + 36) x - 2 x^{1 + 2}}{{(x^{2} + 36)}^{2}}$

Step 2.1.6

Add $1$ and $2$ .

$7 \frac{2 (x^{2} + 36) x - 2 x^{3}}{{(x^{2} + 36)}^{2}}$

Step 2.1.7

Combine $7$ and $\frac{2 (x^{2} + 36) x - 2 x^{3}}{{(x^{2} + 36)}^{2}}$ .

$\frac{7 (2 (x^{2} + 36) x - 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Apply the distributive property.

$\frac{7 ((2 x^{2} + 2 \cdot 36) x - 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.2

Apply the distributive property.

$\frac{7 (2 x^{2} x + 2 \cdot 36 x - 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.3

Apply the distributive property.

$\frac{7 (2 x^{2} x) + 7 (2 \cdot 36 x) + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4

Simplify the numerator.

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Step 2.1.8.4.1

Simplify each term.

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Step 2.1.8.4.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.8.4.1.1.1

Move $x$ .

$\frac{7 (2 (x \cdot x^{2})) + 7 (2 \cdot 36 x) + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.1.1.2

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

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Step 2.1.8.4.1.1.2.1

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

$\frac{7 (2 (x^{1} x^{2})) + 7 (2 \cdot 36 x) + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.1.1.2.2

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

$\frac{7 (2 x^{1 + 2}) + 7 (2 \cdot 36 x) + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.1.1.3

Add $1$ and $2$ .

$\frac{7 (2 x^{3}) + 7 (2 \cdot 36 x) + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.1.2

Multiply $2$ by $7$ .

$\frac{14 x^{3} + 7 (2 \cdot 36 x) + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.1.3

Multiply $2$ by $36$ .

$\frac{14 x^{3} + 7 (72 x) + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.1.4

Multiply $72$ by $7$ .

$\frac{14 x^{3} + 504 x + 7 (- 2 x^{3})}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.1.5

Multiply $- 2$ by $7$ .

$\frac{14 x^{3} + 504 x - 14 x^{3}}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.2

Combine the opposite terms in $14 x^{3} + 504 x - 14 x^{3}$ .

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Step 2.1.8.4.2.1

Subtract $14 x^{3}$ from $14 x^{3}$ .

$\frac{504 x + 0}{{(x^{2} + 36)}^{2}}$

Step 2.1.8.4.2.2

Add $504 x$ and $0$ .

$f' (x) = \frac{504 x}{{(x^{2} + 36)}^{2}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{504 x}{{(x^{2} + 36)}^{2}}$ .

$\frac{504 x}{{(x^{2} + 36)}^{2}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $\frac{504 x}{{(x^{2} + 36)}^{2}} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$\frac{504 x}{{(x^{2} + 36)}^{2}} = 0$

Step 3.2

Set the numerator equal to zero.

$504 x = 0$

Step 3.3

Divide each term in $504 x = 0$ by $504$ and simplify.

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Step 3.3.1

Divide each term in $504 x = 0$ by $504$ .

$\frac{504 x}{504} = \frac{0}{504}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $504$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{504 x}{504} = \frac{0}{504}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{504}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $504$ .

$x = 0$

Step 4

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 5

After finding the point that makes the derivative $f' (x) = \frac{504 x}{{(x^{2} + 36)}^{2}}$ equal to $0$ or undefined, the interval to check where $f (x) = \frac{7 x^{2}}{x^{2} + 36}$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

$(- \infty, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ 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.

$f' (- 1) = \frac{504 (- 1)}{{({(- 1)}^{2} + 36)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $504$ by $- 1$ .

$f' (- 1) = \frac{- 504}{{({(- 1)}^{2} + 36)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$f' (- 1) = \frac{- 504}{{(1 + 36)}^{2}}$

Step 6.2.2.2

Add $1$ and $36$ .

$f' (- 1) = \frac{- 504}{37^{2}}$

Step 6.2.2.3

Raise $37$ to the power of $2$ .

$f' (- 1) = \frac{- 504}{1369}$

Step 6.2.3

Move the negative in front of the fraction.

$f' (- 1) = - \frac{504}{1369}$

Step 6.2.4

The final answer is $- \frac{504}{1369}$ .

$- \frac{504}{1369}$

Step 6.3

At $x = - 1$ the derivative is $- \frac{504}{1369}$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

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

Step 7

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

$f' (1) = \frac{504 (1)}{{({(1)}^{2} + 36)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $504$ by $1$ .

$f' (1) = \frac{504}{{(1^{2} + 36)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

One to any power is one.

$f' (1) = \frac{504}{{(1 + 36)}^{2}}$

Step 7.2.2.2

Add $1$ and $36$ .

$f' (1) = \frac{504}{37^{2}}$

Step 7.2.2.3

Raise $37$ to the power of $2$ .

$f' (1) = \frac{504}{1369}$

Step 7.2.3

The final answer is $\frac{504}{1369}$ .

$\frac{504}{1369}$

Step 7.3

At $x = 1$ the derivative is $\frac{504}{1369}$ . Since this is positive, the function is increasing on $(0, \infty)$ .

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

Step 8

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

Increasing on: $(0, \infty)$

Decreasing on: $(- \infty, 0)$

Step 9