3.7 - Implicit Differentiation

A process that allows you to take the derivative of multi-variable functions.

• It’s basically just the chain rule, but you have to keep in mind that the derivative of $y$ with respect to $x$ is simply $\frac{dy}{dx}$ .

Example

What is the derivative of $3y^2$ with respect to $x$?

• Use the chain rule - derivative of outside @ inside * derivative of inside. This becomes $6y^1\ast \frac{dy}{dx}$.

Second Example

Find the derivative of $x^2+y^2=4$

• This becomes $2x+2y\ast \frac{dy}{dx}=0$.
• Solving for $\frac{dy}{dx}$ gives us that $\frac{dy}{dx}=-\frac{x}{y}$.

Third Example

What is the derivative of $x^2-xy+y^2=7$?

• This turns into $2x-(x\ast \frac{dy}{dx}+y\ast 1)+2y\ast \frac{dy}{dx}=0$
  • We used the sum rule, the product rule, and the chain rule.
• Eventually we get that $\frac{dy}{dx}=\frac{-2x+y}{-x+2y}$.