Řešení - Derivative

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

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

Vysvětlení krok za krokem

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

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

Decomposing the function for the chain rule.

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

Computing the derivative of a natural logarithm function.

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

Substituting the variable back into the function.

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

Computing the derivative of a cosine function.

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

Simplifying the arithmetic expressions.

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

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