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**Implicit Differentiation**

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

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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.

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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

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**Explicit vs. Implicit We say that the first form of the equation**

defines y implicitly as a function of x.

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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.

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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.

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**Explicit vs. Implicit Definition 3.1.1**

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.

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**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.

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**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:

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**Implicit Differentiation**

We can also take the derivative of the given function without solving for x by using a technique called implicit differentiation.

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**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.

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**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.

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**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.

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**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.

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Example 2 Use implicit differentiation to find dy/dx if

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Example 2 Use implicit differentiation to find dy/dx if

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Example 2 Use implicit differentiation to find dy/dx if

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Example 3 Use implicit differentiation to find if

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Example 3 Use implicit differentiation to find if

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Example 3 Use implicit differentiation to find if

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Example 3 Use implicit differentiation to find if

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Example 3 Use implicit differentiation to find if

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Example 3 Use implicit differentiation to find if

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**Example 4 Find the slopes of the tangent lines to the curve**

at the points (2, -1) and (2, 1).

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**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.

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**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.

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**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.

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

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

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

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Homework Section 3.1 Page 190 1-19 odd 19 (just use implicit)

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