MTH 251 – Differential Calculus

Chapter 3 – Differentiation

Section 3.6

Implicit Differentiation

Equations of Curves

• Explicit: y = f(x) Set of ordered pairs (x, y) = (x, f(x)) 2nd coordinate is given in terms of an expression

involving the 1st coordinate.

• Parametric: x = f(t), y = g(t) Set of ordered pairs (x, y) = (f(t), g(t)) 1st coordinate is given in terms of an expression

of the parameter, t 2nd coordinate is given in terms of an expression

of the parameter, t

• Implicit: f(x,y) = 0 Set of ordered pairs (x,y) such that f(x,y) = 0 f(x,y) is an expression involving x and/or y

Equations of Curves - Example

• A Circle of Radius 2

Explicit:

Parametric:

Implicit:

• Line with slope 2/3 containing the point (0, 5)

Explicit:

Parametric:

Implicit:

24y x

2cos , 2sin , 0 2x t y t t

2 2 4x y

23 5y x

3 , 2 5, x t y t t

2 3 15x y

Explicit Parametric

• If y = f(x) … Let x = g(t) … any expression of t Substitute to get y = f(x) = f(g(t)) Determine the domain for t.

• Example …

2 5y x

AKA: Parameterization

Parametric Explicit

• If x = f(t), y = g(t) … Solve x = f(t) for t, giving t = h(x) Substitute to get y = g(t) = g(h(x)) Determine the domain for x.

• Example …

2 3, 1x t y t

AKA: Eliminating the Parameter

Explicit Implicit

• If y = f(x) … Move everything to one side of the equation. (optional) Simplify. I.E. f(x) – y = 0 or y – f(x) = 0

• Example …

2 5y x

Implicit Explicit

• If f(x, y) = 0 … Solve for y ... if possible!

• Examples …

2 2 0xy x

2 2 0y x x y

cos sin 0x y y x

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• i.e. use the chain rule and

2. Algebraically solve for dy/dx

d dyy

dx dx

Implicit Differentiation - Examples

2 2x y xy

2cos y x y

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 y2 – 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 1st derivative into the 2nd derivative

to get the result in terms of x and y only.

• Higher Order Derivatives … continue likewise!

• Example: Find the 1st & 2nd derivatives of …2 2 25x y

Power Rule … one more time

• What if n is a fraction?

1, ifn ndx nx n Z

dx

pqnd d

x xdx dx

pn

q

pqy x q py x

q pd dy x

dx dx

1 1q pdyqy px

dx

1 1

1 111

pq

pq

p pn

qq

dy p x p x px nx

dx q y q qx

if n Q