Implicit Differentiation and Related Rates

On this page we hope to demonstrate the following:

How to use Implicit Differentiation to find derivatives for difficult functions
Ex) Find dy/dx of y² + 1 + 10xy = 4x² when x = 0.

How to find rates of change for related variables (related rates problems)
Ex) Find the rate at which the volume of a sphere is changing when the radius is 5.

Implicit Differentiation

Often, we cannot solve explicitly for y when we are given the equation of a curve. For example, suppose we had the curve e^y + y²x + x² = sin(y), and we needed to find the tangent line. Normally, we would solve for dy/dx to find the slope. Our current method for that requires us to solve for y explicitly, however (this means to get y by itself, i.e. y = f(x)). When we cannot differentiate explicitly, we differentiate implicitly.

Implicit Differentiation is differentiating an equation without explicitly solving for what we want to find before the differentiating. We go through an equation term by term and differentiate with respect to a variable, being careful to use the chain rule when necessary. When we differentiate implicitly, we end up with a complicated equation that contains derivatives, like dy/dx, as well as variables, like y and x. It may be easiest to see this in an example.

Example 1: Find the tangential slope of the curve sin(y) + y + x = 1 at y = π/2

First, let's look at a graph of this curve.