Calculus Examples

$y = \frac{2}{2 x^{4} - 5}$

Step 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 $f (x) = 2$ and $g (x) = 2 x^{4} - 5$ .

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

Step 2

Differentiate.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(2 x^{4} - 5) \cdot 0 - 1 \cdot 2 \frac{d}{d x} [2 x^{4} - 5]}{{(2 x^{4} - 5)}^{2}}$

Step 2.2

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

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

Step 3

Evaluate $\frac{d}{d x} [2 x^{4}]$ .

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Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{(2 x^{4} - 5) \cdot 0 - 1 \cdot 2 (2 (4 x^{3}) + \frac{d}{d x} [- 5])}{{(2 x^{4} - 5)}^{2}}$

Step 3.3

Multiply $4$ by $2$ .

$\frac{(2 x^{4} - 5) \cdot 0 - 1 \cdot 2 (8 x^{3} + \frac{d}{d x} [- 5])}{{(2 x^{4} - 5)}^{2}}$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$\frac{(2 x^{4} - 5) \cdot 0 - 1 \cdot 2 (8 x^{3} + 0)}{{(2 x^{4} - 5)}^{2}}$

Step 4.2

Add $8 x^{3}$ and $0$ .

$\frac{(2 x^{4} - 5) \cdot 0 - 1 \cdot 2 (8 x^{3})}{{(2 x^{4} - 5)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{2 x^{4} \cdot 0 - 5 \cdot 0 - 1 \cdot 2 (8 x^{3})}{{(2 x^{4} - 5)}^{2}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $2 x^{4} \cdot 0$ .

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Step 5.2.1.1.1

Multiply $0$ by $2$ .

$\frac{0 x^{4} - 5 \cdot 0 - 1 \cdot 2 (8 x^{3})}{{(2 x^{4} - 5)}^{2}}$

Step 5.2.1.1.2

Multiply $0$ by $x^{4}$ .

$\frac{0 - 5 \cdot 0 - 1 \cdot 2 (8 x^{3})}{{(2 x^{4} - 5)}^{2}}$

Step 5.2.1.2

Multiply $- 5$ by $0$ .

$\frac{0 + 0 - 1 \cdot 2 (8 x^{3})}{{(2 x^{4} - 5)}^{2}}$

Step 5.2.1.3

Multiply $- 1$ by $2$ .

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

Step 5.2.1.4

Multiply $8$ by $- 2$ .

$\frac{0 + 0 - 16 x^{3}}{{(2 x^{4} - 5)}^{2}}$

Step 5.2.2

Combine the opposite terms in $0 + 0 - 16 x^{3}$ .

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Step 5.2.2.1

Add $0$ and $0$ .

$\frac{0 - 16 x^{3}}{{(2 x^{4} - 5)}^{2}}$

Step 5.2.2.2

Subtract $16 x^{3}$ from $0$ .

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

Step 5.3

Move the negative in front of the fraction.

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