Question

Помогите, формулы дифферинцирования

Answer 1

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Answer 2

Formulas for Differentiation

The formulas for differentiation are essential in calculus and are used to find the derivative of a function with respect to a given variable.

The general formula for differentiation is:

$$ \frac{d}{dx}f(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} $$

This formula represents the rate of change of a function with respect to the variable x.

Specific Differentiation Formulas

1. Logistic Function Differentiation: - The logistic function is commonly used in machine learning and statistics. The derivative of the logistic function is given by: - $$ \frac{d \log \sigma(z)}{d z} = \sigma(-z) $$ - $$ \frac{d \log \sigma(-z)}{d z} = -\sigma(z) $$ - These formulas are used to find the derivative of the logistic function with respect to the input variable.

Source: 2. Trigonometric Function Differentiation: - Differentiation of trigonometric functions involves finding the derivative of functions such as sine, cosine, and tangent. For example: - $$ \frac{d}{dx}(\sin x) = \cos x $$ - $$ \frac{d}{dx}(\cos x) = -\sin x $$ - These formulas are used to find the derivative of trigonometric functions.

Source: 3. General Differentiation Rules: - In addition to specific function differentiations, there are general rules for differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules provide a systematic way to find the derivative of various functions.

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Conclusion

Differentiation is a fundamental concept in calculus and is used to analyze the rate of change of functions. The specific formulas mentioned above are essential tools for finding the derivatives of common functions encountered in mathematics, engineering, and other fields.