Calculus Examples

$f (x) = \frac{c^{8} - x^{8}}{c^{8} + x^{8}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d c} [\frac{f (c)}{g (c)}]$ is $\frac{g (c) \frac{d}{d c} [f (c)] - f (c) \frac{d}{d c} [g (c)]}{{g (c)}^{2}}$ where $f (c) = c^{8} - x^{8}$ and $g (c) = c^{8} + x^{8}$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $c^{8} - x^{8}$ with respect to $c$ is $\frac{d}{d c} [c^{8}] + \frac{d}{d c} [- x^{8}]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d c} [c^{n}]$ is $n c^{n - 1}$ where $n = 8$ .

$\frac{(c^{8} + x^{8}) (8 c^{7} + \frac{d}{d c} [- x^{8}]) - (c^{8} - x^{8}) \frac{d}{d c} [c^{8} + x^{8}]}{{(c^{8} + x^{8})}^{2}}$

Step 2.3

Since $- x^{8}$ is constant with respect to $c$ , the derivative of $- x^{8}$ with respect to $c$ is $0$ .

$\frac{(c^{8} + x^{8}) (8 c^{7} + 0) - (c^{8} - x^{8}) \frac{d}{d c} [c^{8} + x^{8}]}{{(c^{8} + x^{8})}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $8 c^{7}$ and $0$ .

$\frac{(c^{8} + x^{8}) (8 c^{7}) - (c^{8} - x^{8}) \frac{d}{d c} [c^{8} + x^{8}]}{{(c^{8} + x^{8})}^{2}}$

Step 2.4.2

Move $8$ to the left of $c^{8} + x^{8}$ .

$\frac{8 \cdot (c^{8} + x^{8}) c^{7} - (c^{8} - x^{8}) \frac{d}{d c} [c^{8} + x^{8}]}{{(c^{8} + x^{8})}^{2}}$

Step 2.5

By the Sum Rule, the derivative of $c^{8} + x^{8}$ with respect to $c$ is $\frac{d}{d c} [c^{8}] + \frac{d}{d c} [x^{8}]$ .

$\frac{8 (c^{8} + x^{8}) c^{7} - (c^{8} - x^{8}) (\frac{d}{d c} [c^{8}] + \frac{d}{d c} [x^{8}])}{{(c^{8} + x^{8})}^{2}}$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d c} [c^{n}]$ is $n c^{n - 1}$ where $n = 8$ .

$\frac{8 (c^{8} + x^{8}) c^{7} - (c^{8} - x^{8}) (8 c^{7} + \frac{d}{d c} [x^{8}])}{{(c^{8} + x^{8})}^{2}}$

Step 2.7

Since $x^{8}$ is constant with respect to $c$ , the derivative of $x^{8}$ with respect to $c$ is $0$ .

$\frac{8 (c^{8} + x^{8}) c^{7} - (c^{8} - x^{8}) (8 c^{7} + 0)}{{(c^{8} + x^{8})}^{2}}$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $8 c^{7}$ and $0$ .

$\frac{8 (c^{8} + x^{8}) c^{7} - (c^{8} - x^{8}) (8 c^{7})}{{(c^{8} + x^{8})}^{2}}$

Step 2.8.2

Multiply $8$ by $- 1$ .

$\frac{8 (c^{8} + x^{8}) c^{7} - 8 (c^{8} - x^{8}) c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(8 c^{8} + 8 x^{8}) c^{7} - 8 (c^{8} - x^{8}) c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{8 c^{8} c^{7} + 8 x^{8} c^{7} - 8 (c^{8} - x^{8}) c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{8 c^{8} c^{7} + 8 x^{8} c^{7} + (- 8 c^{8} - 8 (- x^{8})) c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{8 c^{8} c^{7} + 8 x^{8} c^{7} - 8 c^{8} c^{7} - 8 (- x^{8}) c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Combine the opposite terms in $8 c^{8} c^{7} + 8 x^{8} c^{7} - 8 c^{8} c^{7} - 8 (- x^{8}) c^{7}$ .

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Step 3.5.1.1

Subtract $8 c^{8} c^{7}$ from $8 c^{8} c^{7}$ .

$\frac{8 x^{8} c^{7} + 0 - 8 (- x^{8}) c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3.5.1.2

Add $8 x^{8} c^{7}$ and $0$ .

$\frac{8 x^{8} c^{7} - 8 (- x^{8}) c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3.5.2

Multiply $- 1$ by $- 8$ .

$\frac{8 x^{8} c^{7} + 8 x^{8} c^{7}}{{(c^{8} + x^{8})}^{2}}$

Step 3.5.3

Add $8 x^{8} c^{7}$ and $8 x^{8} c^{7}$ .

$\frac{16 x^{8} c^{7}}{{(c^{8} + x^{8})}^{2}}$