Calculus Examples

$\frac{2 x}{z^{2} - y^{2}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d z} [\frac{f (z)}{g (z)}]$ is $\frac{g (z) \frac{d}{d z} [f (z)] - f (z) \frac{d}{d z} [g (z)]}{{g (z)}^{2}}$ where $f (z) = 2 x$ and $g (z) = z^{2} - y^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Since $- y^{2}$ is constant with respect to $z$ , the derivative of $- y^{2}$ with respect to $z$ is $0$ .

$\frac{(z^{2} - y^{2}) \cdot 0 - 1 \cdot 2 x (2 z + 0)}{{(z^{2} - y^{2})}^{2}}$

Step 2.5

Add $2 z$ and $0$ .

$\frac{(z^{2} - y^{2}) \cdot 0 - 1 \cdot 2 x (2 z)}{{(z^{2} - y^{2})}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{z^{2} \cdot 0 - y^{2} \cdot 0 - 1 \cdot 2 x (2 z)}{{(z^{2} - y^{2})}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $z^{2}$ by $0$ .

$\frac{0 - y^{2} \cdot 0 - 1 \cdot 2 x (2 z)}{{(z^{2} - y^{2})}^{2}}$

Step 3.2.1.2

Multiply $- y^{2} \cdot 0$ .

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Step 3.2.1.2.1

Multiply $0$ by $- 1$ .

$\frac{0 + 0 y^{2} - 1 \cdot 2 x (2 z)}{{(z^{2} - y^{2})}^{2}}$

Step 3.2.1.2.2

Multiply $0$ by $y^{2}$ .

$\frac{0 + 0 - 1 \cdot 2 x (2 z)}{{(z^{2} - y^{2})}^{2}}$

Step 3.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{0 + 0 - 1 \cdot 2 \cdot 2 x z}{{(z^{2} - y^{2})}^{2}}$

Step 3.2.1.4

Multiply $- 1 \cdot 2 \cdot 2$ .

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Step 3.2.1.4.1

Multiply $- 1$ by $2$ .

$\frac{0 + 0 - 2 \cdot 2 x z}{{(z^{2} - y^{2})}^{2}}$

Step 3.2.1.4.2

Multiply $- 2$ by $2$ .

$\frac{0 + 0 - 4 x z}{{(z^{2} - y^{2})}^{2}}$

Step 3.2.2

Combine the opposite terms in $0 + 0 - 4 x z$ .

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Step 3.2.2.1

Add $0$ and $0$ .

$\frac{0 - 4 x z}{{(z^{2} - y^{2})}^{2}}$

Step 3.2.2.2

Subtract $4 x z$ from $0$ .

$\frac{- 4 x z}{{(z^{2} - y^{2})}^{2}}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{4 x z}{{(z^{2} - y^{2})}^{2}}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = y$ .

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

Step 3.4.2

Apply the product rule to $(z + y) (z - y)$ .

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