Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, rather than explicitly as y = f(x).

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

To differentiate implicitly, take the derivative of both sides of the equation with respect to x, applying the chain rule to terms involving y, and then solve for dy/dx.

The derivative is $$n y^{n-1} \frac{dy}{dx}$$ , where n is a constant.

The chain rule states that if a variable y depends on u, which in turn depends on x, then the derivative of y with respect to x is $$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$$ .

$$\frac{dy}{dx} = -\frac{8x^3y^2}{4x^4y - 3y^2}$$ .

$$\frac{dy}{dx} = -\frac{1}{4}$$ .

$$\frac{dy}{dx} = -\frac{1}{5}$$ .

The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is $$\frac{d}{dx}(uv) = u'v + uv'$$ .

$$\frac{dy}{dx} = \frac{2y(x - y^2)}{x(6y^2 - x)}$$ .