Calculus Examples

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

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 $f (x) = 2 x^{2} + 3$ and $g (x) = 4 x^{2} + 5$ .

$\frac{(4 x^{2} + 5) \frac{d}{d x} [2 x^{2} + 3] - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{(4 x^{2} + 5) (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3]) - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.2

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

$\frac{(4 x^{2} + 5) (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]) - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

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 $n = 2$ .

$\frac{(4 x^{2} + 5) (2 (2 x) + \frac{d}{d x} [3]) - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.4

Multiply $2$ by $2$ .

$\frac{(4 x^{2} + 5) (4 x + \frac{d}{d x} [3]) - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.5

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

$\frac{(4 x^{2} + 5) (4 x + 0) - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $4 x$ and $0$ .

$\frac{(4 x^{2} + 5) (4 x) - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.6.2

Move $4$ to the left of $4 x^{2} + 5$ .

$\frac{4 \cdot (4 x^{2} + 5) x - (2 x^{2} + 3) \frac{d}{d x} [4 x^{2} + 5]}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.7

By the Sum Rule, the derivative of $4 x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [5]$ .

$\frac{4 (4 x^{2} + 5) x - (2 x^{2} + 3) (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [5])}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.8

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\frac{4 (4 x^{2} + 5) x - (2 x^{2} + 3) (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5])}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.9

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

$\frac{4 (4 x^{2} + 5) x - (2 x^{2} + 3) (4 (2 x) + \frac{d}{d x} [5])}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.10

Multiply $2$ by $4$ .

$\frac{4 (4 x^{2} + 5) x - (2 x^{2} + 3) (8 x + \frac{d}{d x} [5])}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.11

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{4 (4 x^{2} + 5) x - (2 x^{2} + 3) (8 x + 0)}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.12

Simplify the expression.

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Step 1.1.2.12.1

Add $8 x$ and $0$ .

$\frac{4 (4 x^{2} + 5) x - (2 x^{2} + 3) (8 x)}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.2.12.2

Multiply $8$ by $- 1$ .

$\frac{4 (4 x^{2} + 5) x - 8 (2 x^{2} + 3) x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{(4 (4 x^{2}) + 4 \cdot 5) x - 8 (2 x^{2} + 3) x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$\frac{4 (4 x^{2}) x + 4 \cdot 5 x - 8 (2 x^{2} + 3) x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.3

Apply the distributive property.

$\frac{4 (4 x^{2}) x + 4 \cdot 5 x + (- 8 (2 x^{2}) - 8 \cdot 3) x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.4

Apply the distributive property.

$\frac{4 (4 x^{2}) x + 4 \cdot 5 x - 8 (2 x^{2}) x - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5

Simplify the numerator.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

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

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Step 1.1.3.5.1.1.1

Move $x$ .

$\frac{4 (4 (x \cdot x^{2})) + 4 \cdot 5 x - 8 (2 x^{2}) x - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.1.2

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

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Step 1.1.3.5.1.1.2.1

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

$\frac{4 (4 (x^{1} x^{2})) + 4 \cdot 5 x - 8 (2 x^{2}) x - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.1.2.2

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

$\frac{4 (4 x^{1 + 2}) + 4 \cdot 5 x - 8 (2 x^{2}) x - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.1.3

Add $1$ and $2$ .

$\frac{4 (4 x^{3}) + 4 \cdot 5 x - 8 (2 x^{2}) x - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.2

Multiply $4$ by $4$ .

$\frac{16 x^{3} + 4 \cdot 5 x - 8 (2 x^{2}) x - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.3

Multiply $4$ by $5$ .

$\frac{16 x^{3} + 20 x - 8 (2 x^{2}) x - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.4

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

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Step 1.1.3.5.1.4.1

Move $x$ .

$\frac{16 x^{3} + 20 x - 8 (2 (x \cdot x^{2})) - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.4.2

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

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Step 1.1.3.5.1.4.2.1

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

$\frac{16 x^{3} + 20 x - 8 (2 (x^{1} x^{2})) - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.4.2.2

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

$\frac{16 x^{3} + 20 x - 8 (2 x^{1 + 2}) - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.4.3

Add $1$ and $2$ .

$\frac{16 x^{3} + 20 x - 8 (2 x^{3}) - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.5

Multiply $2$ by $- 8$ .

$\frac{16 x^{3} + 20 x - 16 x^{3} - 8 \cdot 3 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.1.6

Multiply $- 8$ by $3$ .

$\frac{16 x^{3} + 20 x - 16 x^{3} - 24 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.2

Combine the opposite terms in $16 x^{3} + 20 x - 16 x^{3} - 24 x$ .

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Step 1.1.3.5.2.1

Subtract $16 x^{3}$ from $16 x^{3}$ .

$\frac{20 x + 0 - 24 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.2.2

Add $20 x$ and $0$ .

$\frac{20 x - 24 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.5.3

Subtract $24 x$ from $20 x$ .

$\frac{- 4 x}{{(4 x^{2} + 5)}^{2}}$

Step 1.1.3.6

Move the negative in front of the fraction.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4 x}{{(4 x^{2} + 5)}^{2}}$ .

$- \frac{4 x}{{(4 x^{2} + 5)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4 x}{{(4 x^{2} + 5)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{4 x}{{(4 x^{2} + 5)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 x = 0$

Step 2.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 2.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $4$ .

$x = 0$

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 $\frac{2 x^{2} + 3}{4 x^{2} + 5}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{2 {(0)}^{2} + 3}{4 {(0)}^{2} + 5}$

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

Raising $0$ to any positive power yields $0$ .

$\frac{2 \cdot 0 + 3}{4 \cdot 0^{2} + 5}$

Step 4.1.2.1.2

Multiply $2$ by $0$ .

$\frac{0 + 3}{4 \cdot 0^{2} + 5}$

Step 4.1.2.1.3

Add $0$ and $3$ .

$\frac{3}{4 \cdot 0^{2} + 5}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{3}{4 \cdot 0 + 5}$

Step 4.1.2.2.2

Multiply $4$ by $0$ .

$\frac{3}{0 + 5}$

Step 4.1.2.2.3

Add $0$ and $5$ .

$\frac{3}{5}$

Step 4.2

List all of the points.

$(0, \frac{3}{5})$

Step 5