Aim: What Is Implicit Differentiation and How Does It Work?

Aim: What Is Implicit Differentiation and How Does It Work? Implicit Do Now: Explicit

Explicit Form Implicit Form Often you can solve for y in term of x Implicit vs. Explicit variable y is written as a function of x derivative of y? Not Always! Implicit Differentiation is used

variables agree un u’ nun-1 variables disagree Chain Rule Differentiating with Respect to x Use Simple Power Rule Use Chain Rule Product Rule Chain Rule Simplify

COMMON ERROR! DON’T FORGET variables agree un u’ nun-1 variables disagree Chain Rule Differentiating with Respect to x Use Simple Power Rule Use Chain Rule Product Rule Chain Rule Simplify

Guidelines for Implicit Differentiation • Differentiate both sides of the equation with respect to x. • Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation. • Factor dy/dx out of the left side of the equation. • Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx.

Model Problem Find dy/dx given y3 + y2 – 5y – x2 = -4 • Differentiate both sides of the equation with respect to x. 2. Collect all terms involving dy/dx on the left side of the equation

(1, 1) slope at (1, 1) (2, 0) slope at (2, 0) (1, -3) slope at (1, -3) y3 + y2 – 5y – x2 = -4 Model Problem Find dy/dx given y3 + y2 – 5y – x2 = -4 3. Factor dy/dx out of the left side of the equation. 4. Solve for dy/dx by dividing by (3y2 + 2y – 5) function? NO und -4/5 1/8

Functions from Equations If a segment of a graph can be represented by a differentiable function, dy/dx will have meaning as the slope. function? NO YES YES Recall: a function is not differentiable at points with vertical tangents nor at points where the function is not continuous

Aim: What Is Implicit Differentiation and How Does It Work? Do Now: Determine the slope of the tangent line to the graph x2 + 4y2 = 4 at the point .

Slope of tangent at is 1/2 Model Problem Determine the slope of the tangent line to the graph x2 + 4y2 = 4 at the point . implicit differentiation solve for dy/dx evaluate for the point

Model Problem Determine the slope of the tangent line to the graph 3(x2 + y2)2 = 100xy and the point (3, 1). Constant and General Power Rules FOIL and isolate dy/dx

Model Problem Determine the slope of the tangent line to the graph 3(x2 + y2)2 = 100xy and the point (3, 1).

Model Problem Determine the slope of the tangent line to the graph 3(x2 + y2)2 = 100xy and the point (3, 1). substitute (3, 1)

find first derivative implicitly: Finding the 2nd Derivative Implicitly Given x2 + y2 = 25, find quotient rule sub –x/y for dy/dx sub 25 for x2+y2

Model Problem Find the tangent line to the graph given by x2(x2 + y2) = y2 at the point implicit differentiation

Model Problem Find the tangent line to the graph given by x2(x2 + y2) = y2 at the point substitute = 3 = m point-slope formula for equation (y – y1) = m(x – x1)

Model Problem Find dy/dx implicitly for the equation sin y = x. Then find the largest interval of the form –a < y < a such that y is a differentiable function of x. explicitly: for the interval -/2 < y < /2, we use and substitute the original equation to arrive at

Aim: What Is Implicit Differentiation and How Does It Work?