MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.7 Implicit Differentiation Copyright © 2010 by Ron Wallace, all rights reserved.

Equations of Curves Explicit: y = f(x)  Set of ordered pairs (x, y) = (x, f(x))  2 nd coordinate is given in terms of an expression involving the 1 st coordinate. Implicit: f(x,y) = 0  Set of ordered pairs (x,y) such that f(x,y) = 0 Note: May not be in this form, just not solved for y.  f(x,y) is an expression involving x and/or y  May or may not be algebraically solvable for y  The derivative, dy/dx, is still needed Solution? Implicit Differentiation

Equations of Curves – Example 1 Line with slope 2/3 containing the point (0, 5)  Explicit:  Implicit:

Equations of Curves – Example 2 A Sideways Parabola  Explicit:  Implicit: Note: Actually, the combination of two functions.

Equations of Curves – Example 3 A Circle of Radius 2  Explicit:  Implicit: Note: Actually, the combination of two functions.

Equations of Curves – Example 4 A Lemniscate  Explicit:  Implicit:  Polar: A plane curve generated by the locus of the point at which a variable tangent to a rectangular hyperbola intersects a perpendicular from the origin to the tangent. F1F1 F2F2 It is also true that PF 1  PF 2 = 8/2 = 4 (the distance between F 1 and F 2 ) P

Implicit Differentiation Finding dy/dx for an implicitly defined function without explicitly solving for y.  Note: The result may (will) be in terms of x & y 1.Differentiate both sides of the equation in terms of x, treating y as a function of x That is, let y = f(x) Use the chain rule: Use: 2.Algebraically solve for f’(x) or dy/dx Note: Replace any occurrence of f(x) with y.

Implicit Differentiation - Examples Compare to differentiating 1 of 2

Implicit Differentiation - Examples 2 of 2

Tangents & Normals Tangent Line  The limit of secant lines.  Slope = dy/dx Normal Line  The line perpendicular to the tangent.  Slope = –1/(dy/dx) Example … find the tangent and normal to the curve y 2 – 2x – 4y – 1 = 0 at the point (–2, 1)

Second Derivatives Implicitly Find the first derivative implicitly. Differentiate the first derivative implicitly.  The answer will be in terms of dy/dx.  Substitute the 1 st derivative into the 2 nd derivative to get the result in terms of x and y only. Higher Order Derivatives … continue likewise!

Second Derivatives Implicitly Example: Find the 1 st & 2 nd derivatives of …

Power Rule … one more time What if n is a fraction?

Power Rule … one more time What if n is a fraction? Let

Differentiating Radicals Using Fractional Exponents