Calculus Examples

$\frac{x^{3}}{\sin (x)}$

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 $f (x) = x^{3}$ and $g (x) = \sin (x)$ .

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3

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

$\frac{\sin (x) (3 x^{2}) - x^{3} \cos (x)}{\sin^{2} (x)}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 \sin (x) x^{2} - x^{3} \cos (x)}{\sin^{2} (x)}$

Step 4.1.2

Reorder factors in $3 \sin (x) x^{2} - x^{3} \cos (x)$ .

$\frac{3 x^{2} \sin (x) - x^{3} \cos (x)}{\sin^{2} (x)}$

Step 4.2

Factor $x^{2}$ out of $3 x^{2} \sin (x) - x^{3} \cos (x)$ .

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Step 4.2.1

Factor $x^{2}$ out of $3 x^{2} \sin (x)$ .

$\frac{x^{2} (3 \sin (x)) - x^{3} \cos (x)}{\sin^{2} (x)}$

Step 4.2.2

Factor $x^{2}$ out of $- x^{3} \cos (x)$ .

$\frac{x^{2} (3 \sin (x)) + x^{2} (- x \cos (x))}{\sin^{2} (x)}$

Step 4.2.3

Factor $x^{2}$ out of $x^{2} (3 \sin (x)) + x^{2} (- x \cos (x))$ .

$\frac{x^{2} (3 \sin (x) - x \cos (x))}{\sin^{2} (x)}$