Implicit Differentiation

Procedure

If an equation defines y implicitly as a differentiable function of x, determine $\frac{dy}{dx}$ as follows:

1. Differentiate both sides of the equation in terms of x.
   1. Use chain rule when differentiating y terms
2. Treat y as a function of x (if you can find that out)
3. Solve for $\frac{dy}{dx}$

Example 1

Certain functions have both variables in a dependent relationship. For example: $x^2+y^2=25$ To find a general derivative, get the derivative of every term $2x+2y=0$ Usually derivative is in terms of x. so apply this derivative for y to find x. $2x+2y\frac{dy}{dx}=0$ $2y\frac{dy}{dx}=-2x$ $\frac{dy}{dx}=\frac{-2x}{2y}$ $\frac{dy}{dx}=-\frac{x}{y}$ This will be the general derivative. So to find at a given point, then you can just sub in x and y. $\frac{d}{dx}(3,2)$ $=-\frac{-3}{2}$ $=1.5$

Example 2

$2xy-y^3=4$ $2x(y)-y^3=4$ Now derive it $2y+2x\frac{d}{dx}y-3y^2\frac{d}{dx}=0$ $2x\frac{d}{dx}y-3y^2\frac{d}{dx}=-2y$