Calculus Examples

$y = \frac{1 - 2 x^{2}}{2 x^{2} - 5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1 - 2 x^{2}}{2 x^{2} - 5})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = 1 - 2 x^{2}$ and $g (x) = 2 x^{2} - 5$ .

$\frac{(2 x^{2} - 5) \frac{d}{d x} [1 - 2 x^{2}] - (1 - 2 x^{2}) \frac{d}{d x} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

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

$\frac{(2 x^{2} - 5) (\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x^{2}]) - (1 - 2 x^{2}) \frac{d}{d x} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.2

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

$\frac{(2 x^{2} - 5) (0 + \frac{d}{d x} [- 2 x^{2}]) - (1 - 2 x^{2}) \frac{d}{d x} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- 2 x^{2}]$ .

$\frac{(2 x^{2} - 5) \frac{d}{d x} [- 2 x^{2}] - (1 - 2 x^{2}) \frac{d}{d x} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.4

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

$\frac{(2 x^{2} - 5) (- 2 \frac{d}{d x} [x^{2}]) - (1 - 2 x^{2}) \frac{d}{d x} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.5

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

$\frac{(2 x^{2} - 5) (- 2 (2 x)) - (1 - 2 x^{2}) \frac{d}{d x} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.6

Multiply $2$ by $- 2$ .

$\frac{(2 x^{2} - 5) (- 4 x) - (1 - 2 x^{2}) \frac{d}{d x} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.7

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

$\frac{(2 x^{2} - 5) (- 4 x) - (1 - 2 x^{2}) (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.8

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

$\frac{(2 x^{2} - 5) (- 4 x) - (1 - 2 x^{2}) (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.9

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

$\frac{(2 x^{2} - 5) (- 4 x) - (1 - 2 x^{2}) (2 (2 x) + \frac{d}{d x} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.10

Multiply $2$ by $2$ .

$\frac{(2 x^{2} - 5) (- 4 x) - (1 - 2 x^{2}) (4 x + \frac{d}{d x} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.11

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

$\frac{(2 x^{2} - 5) (- 4 x) - (1 - 2 x^{2}) (4 x + 0)}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.12

Simplify the expression.

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Step 3.2.12.1

Add $4 x$ and $0$ .

$\frac{(2 x^{2} - 5) (- 4 x) - (1 - 2 x^{2}) (4 x)}{{(2 x^{2} - 5)}^{2}}$

Step 3.2.12.2

Multiply $4$ by $- 1$ .

$\frac{(2 x^{2} - 5) (- 4 x) - 4 (1 - 2 x^{2}) x}{{(2 x^{2} - 5)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{2 x^{2} (- 4 x) - 5 (- 4 x) - 4 (1 - 2 x^{2}) x}{{(2 x^{2} - 5)}^{2}}$

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Apply the distributive property.

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

Step 3.3.4

Simplify the numerator.

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Step 3.3.4.1

Simplify each term.

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Step 3.3.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.4.1.2

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

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Step 3.3.4.1.2.1

Move $x$ .

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

Step 3.3.4.1.2.2

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

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Step 3.3.4.1.2.2.1

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

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

Step 3.3.4.1.2.2.2

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

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

Step 3.3.4.1.2.3

Add $1$ and $2$ .

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

Step 3.3.4.1.3

Multiply $2$ by $- 4$ .

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

Step 3.3.4.1.4

Multiply $- 4$ by $- 5$ .

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

Step 3.3.4.1.5

Multiply $- 4$ by $1$ .

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

Step 3.3.4.1.6

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

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Step 3.3.4.1.6.1

Move $x$ .

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

Step 3.3.4.1.6.2

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

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Step 3.3.4.1.6.2.1

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

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

Step 3.3.4.1.6.2.2

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

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

Step 3.3.4.1.6.3

Add $1$ and $2$ .

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

Step 3.3.4.1.7

Multiply $- 2$ by $- 4$ .

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

Step 3.3.4.2

Combine the opposite terms in $- 8 x^{3} + 20 x - 4 x + 8 x^{3}$ .

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Step 3.3.4.2.1

Add $- 8 x^{3}$ and $8 x^{3}$ .

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

Step 3.3.4.2.2

Add $20 x - 4 x$ and $0$ .

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

Step 3.3.4.3

Subtract $4 x$ from $20 x$ .

$\frac{16 x}{{(2 x^{2} - 5)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{16 x}{{(2 x^{2} - 5)}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{16 x}{{(2 x^{2} - 5)}^{2}}$