Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.

Explicit vs. Implicit An equation of the form y = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation.

Explicit vs. Implicit An equation of the form y = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation. Sometimes functions are defined by equations in which y is not alone on one side. For example is not of the form y = f(x), but it still defines y as a function of x since it can be rewritten as

Explicit vs. Implicit We say that the first form of the equation defines y implicitly as a function of x.

Explicit vs. Implicit An equation in x and y can implicitly define more than one function in x. This can occur when the graph of the equation fails the vertical line test, so it is not the graph of a function.

Explicit vs. Implicit An equation in x and y can implicitly define more than one function in x. This can occur when the graph of the equation fails the vertical line test, so it is not the graph of a function. For example, if we solve the equation of the circle for y in terms of x, we obtain. This gives us two functions that are defined implicitly.

Explicit vs. Implicit Definition We will say that a given equation in x and y defines the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation.

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.

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.

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.

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.

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.

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 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 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 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 a)Use implicit differentiation to find dy/dx for the equation. b)Find the equation of the tangent line at the point

Example 5 a)Use implicit differentiation to find dy/dx for the equation.

Example 5 a)Use implicit differentiation to find dy/dx for the equation. b)Find the equation of the tangent line at the point

Homework Section 3.1 Page odd 19 (just use implicit)