Řešení - Derivative

\frac{\cos (x)}{\sin (x)}

\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

Vysvětlení krok za krokem

Computing the derivative of a logarithm function using the chain rule.

$\frac{d}{d x} [\ln (\sin (x))] = \frac{1}{\sin (x)} \times \frac{d}{d x} [\sin (x)]$

Decomposing the function for the chain rule.

$\frac{d}{d x} [\ln (\sin (x))] = \frac{d}{d x} [\ln (x)] \times \frac{d}{d x} [\sin (x)]$

Computing the derivative of a natural logarithm function.

$\frac{d}{d x} [\ln (x)] \times \frac{d}{d x} [\sin (x)] = \frac{1}{x} \times \frac{d}{d x} [\sin (x)]$

Substituting the variable back into the function.

$\frac{1}{x} \times \frac{d}{d x} [\sin (x)] = \frac{1}{\sin (x)} \times \frac{d}{d x} [\sin (x)]$

Computing the derivative of a sine function.

$\frac{1}{\sin (x)} \times \frac{d}{d x} [\sin (x)] = \frac{1}{\sin (x)} \times \cos (x)$

Simplifying the arithmetic expressions.

$\frac{1}{\sin (x)} \times \cos (x) = \frac{\cos (x)}{\sin (x)}$

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Derivace je klíčová pro pochopení rychlosti změny, extrémů a tvaru grafu funkcí.