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.

We use implicit differentiation when it is difficult or impossible to solve for y explicitly in terms of x.

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

Differentiate both sides with respect to x, applying the chain rule: $$\frac{d}{dx}[y] = \frac{dy}{dx}$$ .

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 u times the derivative of u with respect to x.

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

$$\frac{dy}{dx} = 6x(x^2 + 1)^2$$ .

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

$$\frac{dy}{dx} = -\frac{7y}{4x}$$ .

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