Question

Найти производную функции

Answer 1

$y= \sqrt[5]{x^2+3x} *tg^22x=(x^2+3x)^{ \frac{1}{5} } *tg^22x$
$y'=((x^2+3x)^{ \frac{1}{5} })' *tg^22x+(x^2+3x)^{ \frac{1}{5} } *(tg^22x)' = \\ =\frac{1}{5} *(x^2+3x)^{ -\frac{4}{5} }* (2x+3)*tg^22x+ \\ +(x^2+3x)^{ \frac{1}{5} }) *2tg2x* ( \frac{1}{cos^22x} )*2= \\ = \frac{(2x+3)tg^22x}{5 \sqrt[5]{(x^2+3x)^4} } + 4\frac{ \sqrt[5]{x^2+3x} *tg2x }{cos^22x}$

Answer 2

Finding the Derivative of a Function

To find the derivative of a function, we need to differentiate it with respect to the independent variable. The derivative represents the rate of change of the function at a given point.

Here are a few examples of finding the derivative of different functions:

1. Example 1: - Function: y = (7 - 24x)^10 - Derivative: y' = -240 * (7 - 24x)^9 2. Example 2: - Function: y = 8x + sin(x) + 5 - Derivative: y' = 8 + cos(x) 3. Example 3: - Function: y = sin(ln(x)) * cos(ln(x)) - Derivative: y' = (sin(ln(x)) * cos(ln(x)))' = (cos(ln(x)) * ln(x) + sin(ln(x)) * (1/x)) [[8]](https://uchi.ru/otvety/questions/nayti-proizvodnuyu-podrobno-y-sin-lnx-cos-lnx)

4. Example 4: - Function: f(x) = x^2 * ln(x) - Derivative: f'(x) = 2x * ln(x) + x Please note that these examples are just a few instances, and there are various rules and techniques for finding derivatives depending on the type of function.