implicit differentiation

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, rather than explicitly solving for y.

To apply implicit differentiation, differentiate both sides of the equation with respect to x, treating y as a function of x. Use the chain rule for terms involving y.

$$\frac{dy}{dx} = \frac{5}{y}$$ .

$$\frac{dy}{dx} = -\frac{y}{x + 2y}$$ .

$$\frac{dy}{dx} = -\frac{x^2}{y^2}$$ .

$$\frac{dy}{dx} = \frac{2x}{3y}$$ .

The chain rule states that if y is a function of x, then the derivative of y with respect to x is the derivative of y with respect to another variable multiplied by the derivative of that variable with respect to x.

Treating y as a function of x allows us to differentiate y with respect to x, which is essential for finding the slope of the tangent line to the curve defined by the implicit equation.

First, find dy/dx using implicit differentiation, then substitute the x and y coordinates of the specific point into the derivative to find the slope.

$$\frac{dy}{dx} = -\frac{x}{y}$$ .