Section 2.5 – Implicit Differentiation

Explicit Equations The functions that we have differentiated and handled so far can be described by expressing one variable explicitly in terms of another variable. For example: Or, in general, y = f(x).

Implicit Equations Some functions, however, are defined implicitly ( not in the form y = f(x) ) by a relation between x and y such as: It is possible to solve some Implicit Equations for y : Yet, it is difficult to rewrite most Implicit Equations explicitly. Thus, we must be introduced to a new technique to differentiate these implicit functions.

*Reminder* Technically the Chain Rule can be applied to every derivative:

Derivatives Involving the Dependent Variable (y) Find the derivative of each expression a. The derivative of y with respect to x is… the derivative of y. This is another way to write y prime. b. The Chain Rule is Required.

Instructions for Implicit Differentiation If y is an equation defined implicitly as a differentiable function of x, to find the derivative: with respect to x 1.Differentiate both sides of the equation with respect to x. (Remember that y is really a function of x for part of the curve and use the Chain Rule when differentiating terms containing y ) 2.Collect all terms involving dy/dx on the left side of the equation, and move the other terms to the right side. 3.Factor dy/dx out of the left side 4.Solve for dy/dx

Example 1 If is a differentiable function of x such that find. Chain Rule Differentiate both sides. Product AND Constant Multiple Rules

Example 2 Find if. Chain Rule Twice Differentiate both sides Product Rule

Example 3 Find if. Chain Rule Find the first derivative by Differentiating both sides. Now Find the Second Derivative Quotient Rule Remember:

More with Derivatives Now evaluate the limits and find tangent lines.

Example 1 Find the slope of a line tangent to the circle at the point. Chain Rule Find the derivative by differentiating both sides. Evaluate the derivative at x=5 and y=4.

Example 2 If and, find. Chain Rule Find the derivative by differentiating both sides.

Example 2 (continued) If and, find. Evaluate the derivative with the given information.

Example 3 Chain Rule Find the derivative by differentiating both sides. Now evaluate the derivative at x=3 and y=4. Find an equation of the tangent to the circle at the point. Use the Point-Slope Formula to find the equation of the tangent line