Question

Найти производную функции: у=arccos x/ корень из 1+2 sin x

Answer 1

Не могу вставить тебе решение, будь другом, перейди на вычисление производных онлайн, я твое выражение уже переформировала в соответствии с их синтаксисом, только забей в форму "у=arccos(x)/Sqrt(1+2sin(x))" - меня очень сильно выручает этот сервис.

Answer 2

Calculating the Derivative of the Function

To find the derivative of the function у = arccos(x) / √(1 + 2sin(x)), we can use the chain rule and the derivative rules for trigonometric functions.

Let's break down the steps:

1. Apply the chain rule to the function arccos(x): - The derivative of arccos(x) is -1 / √(1 - x^2). - Multiply this derivative by the derivative of the inner function, which is 1.

So, the derivative of arccos(x) is -1 / √(1 - x^2).

2. Apply the chain rule to the function √(1 + 2sin(x)): - The derivative of √(1 + 2sin(x)) is (cos(x)) / √(1 + 2sin(x)). - Multiply this derivative by the derivative of the inner function, which is 2cos(x).

So, the derivative of √(1 + 2sin(x)) is 2cos^2(x) / √(1 + 2sin(x)).

3. Combine the derivatives using the quotient rule: - The derivative of у = arccos(x) / √(1 + 2sin(x)) is given by:

(derivative of numerator * denominator - numerator * derivative of denominator) / (denominator^2).

Substituting the derivatives we found earlier, we get:

( -1 / √(1 - x^2) * √(1 + 2sin(x)) - arccos(x) * 2cos^2(x) / √(1 + 2sin(x)) ) / ( √(1 + 2sin(x))^2 ).

Simplifying this expression further is possible, but it may not lead to a significantly simpler form.

Please note that the above derivative is a general expression and can be further simplified depending on the specific value of x.

I hope this helps! Let me know if you have any further questions.