Calculus Examples

$\frac{d}{d x} \cdot \frac{x^{12}}{x^{2}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d \cdot d} [\frac{f (d)}{g (d)}]$ is $\frac{g (d) \frac{d}{d \cdot d} [f (d)] - f (d) \frac{d}{d \cdot d} [g (d)]}{{g (d)}^{2}}$ where $f (d) = x^{12}$ and $g (d) = x^{2}$ .

$\frac{x^{2} \frac{d}{d \cdot d} [x^{12}] - x^{12} \frac{d}{d \cdot d} [x^{2}]}{{(x^{2})}^{2}}$

Step 2

Differentiate using the Constant Rule.

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Step 2.1

Since $x^{12}$ is constant with respect to $d$ , the derivative of $x^{12}$ with respect to $d$ is $0$ .

$\frac{x^{2} \cdot 0 - x^{12} \frac{d}{d \cdot d} [x^{2}]}{{(x^{2})}^{2}}$

Step 2.2

Since $x^{2}$ is constant with respect to $d$ , the derivative of $x^{2}$ with respect to $d$ is $0$ .

$\frac{x^{2} \cdot 0 - x^{12} \cdot 0}{{(x^{2})}^{2}}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

$\frac{0 - x^{12} \cdot 0}{{(x^{2})}^{2}}$

Step 3.1.1.2

Multiply $- x^{12} \cdot 0$ .

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Step 3.1.1.2.1

Multiply $0$ by $- 1$ .

$\frac{0 + 0 x^{12}}{{(x^{2})}^{2}}$

Step 3.1.1.2.2

Multiply $0$ by $x^{12}$ .

$\frac{0 + 0}{{(x^{2})}^{2}}$

Step 3.1.2

Add $0$ and $0$ .

$\frac{0}{{(x^{2})}^{2}}$

Step 3.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{0}{x^{2 \cdot 2}}$

Step 3.2.2

Multiply $2$ by $2$ .

$\frac{0}{x^{4}}$

Step 3.3

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

$0$