Algebra Examples

$\frac{4 \sin (x)}{x^{4}}$

Step 1

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

$4 \frac{d}{d x} [\frac{\sin (x)}{x^{4}}]$

Step 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 $f (x) = \sin (x)$ and $g (x) = x^{4}$ .

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

Step 3

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 3.1

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

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

Step 3.2

Multiply $4$ by $2$ .

$4 \frac{x^{4} \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$4 \frac{x^{4} \cos (x) - \sin (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

$4 \frac{x^{4} \cos (x) - \sin (x) (4 x^{3})}{x^{8}}$

Step 5.2

Simplify with factoring out.

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Step 5.2.1

Multiply $4$ by $- 1$ .

$4 \frac{x^{4} \cos (x) - 4 \sin (x) x^{3}}{x^{8}}$

Step 5.2.2

Factor $x^{3}$ out of $x^{4} \cos (x) - 4 \sin (x) x^{3}$ .

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Step 5.2.2.1

Factor $x^{3}$ out of $x^{4} \cos (x)$ .

$4 \frac{x^{3} (x \cos (x)) - 4 \sin (x) x^{3}}{x^{8}}$

Step 5.2.2.2

Factor $x^{3}$ out of $- 4 \sin (x) x^{3}$ .

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

Step 5.2.2.3

Factor $x^{3}$ out of $x^{3} (x \cos (x)) + x^{3} (- 4 \sin (x))$ .

$4 \frac{x^{3} (x \cos (x) - 4 \sin (x))}{x^{8}}$

Step 6

Cancel the common factors.

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

$4 \frac{x \cos (x) - 4 \sin (x)}{x^{5}}$

Step 7

Combine $4$ and $\frac{x \cos (x) - 4 \sin (x)}{x^{5}}$ .

$\frac{4 (x \cos (x) - 4 \sin (x))}{x^{5}}$