Calculus Examples

$y = \frac{4 x^{3}}{2 x^{2} - 5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{4 x^{3}}{2 x^{2} - 5})$

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

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x^{3}}{2 x^{2} - 5}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x^{3}}{2 x^{2} - 5}]$ .

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

Move $3$ to the left of $2 x^{2} - 5$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Multiply $2$ by $2$ .

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

Step 3.3.7

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

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

Step 3.3.8

Simplify the expression.

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Step 3.3.8.1

Add $4 x$ and $0$ .

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

Step 3.3.8.2

Multiply $4$ by $- 1$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $3$ .

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

Step 3.7

Combine $4$ and $\frac{3 (2 x^{2} - 5) x^{2} - 4 x^{4}}{{(2 x^{2} - 5)}^{2}}$ .

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Apply the distributive property.

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

Step 3.8.4

Simplify the numerator.

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Step 3.8.4.1

Simplify each term.

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Step 3.8.4.1.1

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

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Step 3.8.4.1.1.1

Move $x^{2}$ .

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

Step 3.8.4.1.1.2

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

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

Step 3.8.4.1.1.3

Add $2$ and $2$ .

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

Step 3.8.4.1.2

Multiply $2$ by $3$ .

$\frac{4 (6 x^{4}) + 4 (3 \cdot - 5 x^{2}) + 4 (- 4 x^{4})}{{(2 x^{2} - 5)}^{2}}$

Step 3.8.4.1.3

Multiply $6$ by $4$ .

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

Step 3.8.4.1.4

Multiply $3$ by $- 5$ .

$\frac{24 x^{4} + 4 (- 15 x^{2}) + 4 (- 4 x^{4})}{{(2 x^{2} - 5)}^{2}}$

Step 3.8.4.1.5

Multiply $- 15$ by $4$ .

$\frac{24 x^{4} - 60 x^{2} + 4 (- 4 x^{4})}{{(2 x^{2} - 5)}^{2}}$

Step 3.8.4.1.6

Multiply $- 4$ by $4$ .

$\frac{24 x^{4} - 60 x^{2} - 16 x^{4}}{{(2 x^{2} - 5)}^{2}}$

Step 3.8.4.2

Subtract $16 x^{4}$ from $24 x^{4}$ .

$\frac{8 x^{4} - 60 x^{2}}{{(2 x^{2} - 5)}^{2}}$

Step 3.8.5

Factor $4 x^{2}$ out of $8 x^{4} - 60 x^{2}$ .

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Step 3.8.5.1

Factor $4 x^{2}$ out of $8 x^{4}$ .

$\frac{4 x^{2} (2 x^{2}) - 60 x^{2}}{{(2 x^{2} - 5)}^{2}}$

Step 3.8.5.2

Factor $4 x^{2}$ out of $- 60 x^{2}$ .

$\frac{4 x^{2} (2 x^{2}) + 4 x^{2} (- 15)}{{(2 x^{2} - 5)}^{2}}$

Step 3.8.5.3

Factor $4 x^{2}$ out of $4 x^{2} (2 x^{2}) + 4 x^{2} (- 15)$ .

$\frac{4 x^{2} (2 x^{2} - 15)}{{(2 x^{2} - 5)}^{2}}$

Step 4

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

$y' = \frac{4 x^{2} (2 x^{2} - 15)}{{(2 x^{2} - 5)}^{2}}$

Step 5

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

$\frac{d y}{d x} = \frac{4 x^{2} (2 x^{2} - 15)}{{(2 x^{2} - 5)}^{2}}$