Question

Найти производную неявной функции y(x), заданной уравнением x-y+arctgy=0 подробно

Answer 1

1-y`+y`/(1+y²)=0
1=y`(1-1/(1+y²)
1=y`(1+y²-1)/(1+y²)
1=y`*y²/(1+y²)
y`=1:y²/(1+y²)
y`=(1+y²)/y²

Answer 2

Finding the Derivative of an Implicit Function

To find the derivative of the implicit function \( y(x) \) defined by the equation \( x - y + \arctan(y) = 0 \), we can use implicit differentiation.

The given equation is: \[ x - y + \arctan(y) = 0 \]

To find \( \frac{dy}{dx} \), we differentiate both sides of the equation with respect to \( x \).

Differentiating \( x \) with respect to \( x \) gives \( 1 \). Differentiating \( y \) with respect to \( x \) gives \( \frac{dy}{dx} \). Differentiating \( \arctan(y) \) with respect to \( x \) gives \( \frac{1}{1+y^2} \cdot \frac{dy}{dx} \).

So, the derivative of the implicit function \( y(x) \) is: \[ 1 - \frac{dy}{dx} + \frac{1}{1+y^2} \cdot \frac{dy}{dx} = 0 \]

Solving for \( \frac{dy}{dx} \) will give us the derivative of the implicit function \( y(x) \).

Let's solve for \( \frac{dy}{dx} \).