Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both dependent and independent variables.

We use implicit differentiation when it is difficult or impossible to solve for one variable in terms of another, allowing us to differentiate both sides of an equation with respect to the independent variable.

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

Differentiate both sides: $$x\frac{dy}{dx} + y = 0$$ , leading to $$\frac{dy}{dx} = -\frac{y}{x}$$ .

The Chain Rule states that if a variable depends on another variable, the derivative of the outer function is multiplied by the derivative of the inner function.

The product rule states that if two functions are multiplied, the derivative is given by $$\frac{d}{dx}(uv) = u'v + uv'$$ .

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

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

The implicit function theorem provides conditions under which a relation defines a function implicitly, allowing for the differentiation of variables.

Differentiate $$\frac{dy}{dx}$$ again with respect to x, applying the product and chain rules as necessary.