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

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

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

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

Find dy/dx given y3 + y2 – 5y – x2 = -4 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

Find dy/dx given 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) y3 + y2 – 5y – x2 = -4 function? NO (1, 1) (2, 0) (1, -3) slope at (1, 1) slope at (2, 0) slope at (1, -3) 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 .

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 Slope of tangent at is 1/2

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)

Finding the 2nd Derivative Implicitly Given x2 + y2 = 25, find find first derivative implicitly: 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

point-slope formula for equation (y – y1) = m(x – x1) 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