Calculus Examples

$y = \frac{x^{4}}{4 x^{2} + 4}$

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $4$ .

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 4.2

Add $8 x$ and $0$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 5.2.1.2.1

Move $x^{3}$ .

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

Step 5.2.1.2.2

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

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

Step 5.2.1.2.3

Add $3$ and $2$ .

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

Step 5.2.1.3

Multiply $4$ by $4$ .

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

Step 5.2.1.4

Multiply $4$ by $4$ .

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

Step 5.2.1.5

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.6

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 5.2.1.6.1

Move $x$ .

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

Step 5.2.1.6.2

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

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Step 5.2.1.6.2.1

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

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

Step 5.2.1.6.2.2

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

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

Step 5.2.1.6.3

Add $1$ and $4$ .

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

Step 5.2.1.7

Multiply $- 1$ by $8$ .

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

Step 5.2.2

Subtract $8 x^{5}$ from $16 x^{5}$ .

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

Step 5.3

Factor $8 x^{3}$ out of $8 x^{5} + 16 x^{3}$ .

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Step 5.3.1

Factor $8 x^{3}$ out of $8 x^{5}$ .

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

Step 5.3.2

Factor $8 x^{3}$ out of $16 x^{3}$ .

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

Step 5.3.3

Factor $8 x^{3}$ out of $8 x^{3} (x^{2}) + 8 x^{3} (2)$ .

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

Step 5.4

Simplify the denominator.

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Step 5.4.1

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

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Step 5.4.1.1

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

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

Step 5.4.1.2

Factor $4$ out of $4$ .

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

Step 5.4.1.3

Factor $4$ out of $4 (x^{2}) + 4 (1)$ .

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

Step 5.4.2

Apply the product rule to $4 (x^{2} + 1)$ .

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

Step 5.4.3

Raise $4$ to the power of $2$ .

$\frac{8 x^{3} (x^{2} + 2)}{16 {(x^{2} + 1)}^{2}}$

Step 5.5

Cancel the common factor of $8$ and $16$ .

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Step 5.5.1

Factor $8$ out of $8 x^{3} (x^{2} + 2)$ .

$\frac{8 (x^{3} (x^{2} + 2))}{16 {(x^{2} + 1)}^{2}}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $8$ out of $16 {(x^{2} + 1)}^{2}$ .

$\frac{8 (x^{3} (x^{2} + 2))}{8 (2 {(x^{2} + 1)}^{2})}$

Step 5.5.2.2

Cancel the common factor.

$\frac{8 (x^{3} (x^{2} + 2))}{8 (2 {(x^{2} + 1)}^{2})}$

Step 5.5.2.3

Rewrite the expression.

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