Question

Как найти производную от arctg(x/y)? Объясните как...Как вообще от такого брать производную? От arctg производная 1/(1+x^2),это понятно,а вот с x/y как быть...Вообще объясните мне подробно. Заранее спасибо.Мне нужно найти производную 1 порядка по x, а потом и по y...

Answer 1

Смотри приложенный файл

Answer 2

Когда берем производную от z по x, то у-считается константой(числом).
производная по х=1/y*(1/(1+x^2/y^2))=1/(y*(1+x^2/y^2))  т.е. берешь производную от арктангенса, вместо х у тебя х/у - получится x^2/y^2. потом надо взять производную от х/у- получится 1/у и умножить их.
по у- х-константа.
произв по у=(1/(x^2/y^2+1))*x*(-1/y^2)=(-x)/(y^2*(x^2/y^2+1))

Answer 3

Finding the Derivative of arctg(x/y)

To find the derivative of arctg(x/y), we can use the chain rule of differentiation. The chain rule states that if we have a composite function, such as f(g(x)), then the derivative is f'(g(x)) * g'(x). In this case, we have the composite function arctg(x/y), where the inner function is x/y.

Applying the Chain Rule

To find the derivative of arctg(x/y) with respect to x, we can use the chain rule as follows:

1. Let's denote the function arctg(x/y) as u, and the inner function x/y as v. 2. The derivative of arctg(x/y) with respect to x can be expressed as du/dx = du/dv * dv/dx.

Derivative of arctg(x/y)

The derivative of arctg(x/y) with respect to x is given by the following steps:

1. The derivative of arctg(u) with respect to u is 1/(1+u^2). 2. The derivative of u = arctg(x/y) with respect to v = x/y is -1/(1+(x/y)^2) = -y/(x^2+y^2). 3. The derivative of v = x/y with respect to x is 1/y.

Calculating the Derivative

Now, let's calculate the derivative du/dx using the chain rule:

du/dx = du/dv * dv/dx = (-1/(1+(x/y)^2)) * (1/y) = -y/(x^2+y^2) * (1/y) = -1/(x^2+y^2)

So, the derivative of arctg(x/y) with respect to x is -1/(x^2+y^2).

Derivative with Respect to y

Similarly, to find the derivative of arctg(x/y) with respect to y, we would use the chain rule and obtain the derivative du/dy.

I hope this explanation helps! If you have any further questions or need additional clarification, feel free to ask.