Řešení - Derivative

- e^{x} \sin (e^{x})

- e^{x} \sin{\left(e^{x} \right)}

Vysvětlení krok za krokem

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

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

Decomposing the function for the chain rule.

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

Computing the derivative of a cosine function.

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

Substituting the variable back into the function.

$- \sin (x) \times \frac{d}{d x} [e^{x}] = - \sin (e^{x}) \times \frac{d}{d x} [e^{x}]$

Computing the derivative of an exponential function.

$- \sin (e^{x}) \times \frac{d}{d x} [e^{x}] = - \sin (e^{x}) \times e^{x}$

Simplifying the arithmetic expressions.

$- \sin (e^{x}) \times e^{x} = - e^{x} \sin (e^{x})$

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