Implicit Differentiation,Objective: To find derivatives of functions that we cannot solve for y.,How to do it ?,By now, it should be easy for you to take the derivative of an equation such as,If youre given an equation such as , you can still figure out the derivative by taking the square root of both sides, which gives you in terms of . This is known as finding the derivative explicitly. Its messy, but possiple.,Implicit Differentiation,It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation (but often it is easier to do so).,Find dy/dx for . Can we solve this for y?,Implicit Differentiation,It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation.,For example, we can take the derivative of with the quotient rule:,Implicit Differentiation,We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.,Example 1 Find if,Using implicit differentiation, you get:,Remember that,After you factor out , divide both sides by,Note: Now that you understand that the derivative of an term with repect to will always be multiplied by , and that , we wont write anymore. You should understand that the term is implied.,Example 2 Find if,Using implicit differentiation, you get:,Then simplify:,Next, put all of the terms containing on the left and all of the other terms on the right:,Next, factor out,And isolate :,This can be simplified further to:,Example 3 Find if,Implicit differentiation should result in:,You can simplify this to:,Next, put all of the terms containing on the left and all of the other terms on the right:,Next, factor out,And isolate :,Example 4: Find the derivative of at (2,1).,You need to use implict differentiation to find :,Now, instead of rearranging to isolate , plug in (2,1) immediately and solve for the derivative;,Simplify:,so,Example 5: Find the derivative of at (1,1).,First, cross-multiply :,Take the derivative :,Distribute:,Do not simplify now. Rather, plug in (1,1) right away. This will save you from the algebra:,Now solve for :,Second derivative,Example 6: Find if,Differentiating implicity, you get:,Next, simplify and solve for :,Now its time to take the derivative again:,Finally,substitute for,Problem 1. Find if .,Problem 2. Find if .,Problem3. Find the derivative of each variable with respect to of,Problem4. Find the derivative of each variable with respect to of,Problem 5: Find if,Example 1,Use implicit differentiation to find dy/dx if,Example 2,Use implicit differentiation to find dy/dx if,Example 2,Use implicit differentiation to find dy/dx if,Example 2,Use implicit differentiation to find dy/dx if,Example 3,Use implicit differentiation to find if,Example 3,Use implicit differentiation to find if,Example 3,Use implicit differentiation to find if,Example 3,Use implicit differentiation to find if,Example 3,Use implicit differentiation to find if,Example 3,Use implicit differentiation to find if,Example 3,Use implicit differentiation to find if,Example 4,Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).,Example 4,Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). We know that the slope of the tangent line means the value of the derivative at the given points.,Example 4,Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). We know that the slope of the tangent line means the value of the derivative at the given points.,Example 4,Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). We know that the slope of the tangent line means the value of the derivative at the given points.,Example 5,Use implicit differentiation to find dy/dx for the equation .,