Implicit Differentiation and Derivatives Review

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, without solving for y explicitly.

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

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

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

$$\frac{dy}{dx}$$ represents the rate of change of y with respect to x, or the slope of the tangent line to the curve at a given point.

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 given by $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ .

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

The quotient rule states that if you have a function that is the quotient of two functions u(x) and v(x), then the derivative is given by $$\frac{d(\frac{u}{v})}{dx} = \frac{u'v - uv'}{v^2}$$ .

The derivative at a point gives the slope of the tangent line to the curve at that point, indicating the instantaneous rate of change of the function.

The derivative is $$\frac{dy}{dx} = 2x + 3$$ .