(a) find two explicit functions by solving the equation for y in terms of x,(𝐛) sketch the graph

(a) find two explicit functions by solving the equation for...

Key Concepts

Implicit Differentiation

Implicit differentiation is used when a function is defined implicitly by an equation rather than explicitly by an isolated expression. This technique differentiates both sides of the equation with respect to the independent variable, applying derivative rules to the dependent variable as a function of the independent one. It allows one to find the derivative even when isolating the variable is difficult or results in multiple branches, ensuring that the derivative can be obtained in its general form.

Explicit Differentiation

Explicit differentiation is the process of computing the derivative of a function where the dependent variable is directly expressed in terms of the independent variable. This method uses standard differentiation rules to find the rate of change, making it straightforward when the function has been isolated. It typically involves applying power, root, and constant multiple rules to determine the derivative of each explicit branch.

Chain Rule

The chain rule is a fundamental tool in differentiation that is applied when dealing with composite functions. When the function to differentiate is composed of more than one function nested within each other, the chain rule provides a systematic way to compute the derivative by differentiating the outer function and then multiplying by the derivative of the inner function. This rule is essential in both explicit and implicit differentiation, especially when differentiating functions that involve square roots or other nested operations.

Solving Implicit Equations for Explicit Functions

This concept involves transforming an equation in which the dependent and independent variables are intermixed into one or more functions where the dependent variable is isolated on one side. By doing so, the problem becomes easier to analyze and differentiate, as the variable is expressed explicitly in terms of the other. This is commonly done by algebraically manipulating the original equation, often involving operations like taking square roots, which can lead to multiple branches of solutions.

Graphing Conic Sections

Graphing conic sections entails interpreting algebraic equations representing curves such as ellipses, hyperbolas, and parabolas. In this context, the given equation can be rearranged into a standard form that reveals the structure of the curve, allowing one to sketch its distinct components or branches. Understanding the geometry helps in visualizing the behavior of the function, such as symmetry and orientation, which are essential for a complete analysis of the equation.

Transcript

00:02 All right.

00:03 I'm going to look at the equation 9 y squared minus x squared equals to 9.

00:07 We're looking to go through in this case.

00:09 And the first thing i want to do is to solve this equation for y.

00:13 So the way we would do this is we would add x squared to both sides.

00:18 And we would get 9 y squared equals 9 plus x squared.

00:22 And then i'm going to divide both sides by 9 and i get y squared would be equal to 9 plus x squared over 9.

00:30 And then solving this for y, we would have to take the square root of both sides.

00:35 So if we took the square root of both sides here, this will give me the square root of 9 plus x squared in the numerator.

00:43 And then i'm going to go through and just take the square root of 9 right now.

00:46 It's 3.

00:46 And then since we're taking the square root of that, we need to take plus or minus.

00:51 So this, we end up really getting two equations when we go through and we solve for y.

00:56 Now, if we were to sketch the graphs of these two by hand, it would take a little time.

01:02 And really, i like to write it like this, just a personal preference.

01:05 This is plus or minus one -third on square root of 9 plus x squared is what we get there.

01:12 If we were to sketch the graph of that, i've done it already over here.

01:15 I use my graphing calculator to do this for us.

01:18 This is kind of what we get for the graph of these two.

01:21 We see that it's two different functions.

01:23 We have a top half and we have a bottom half.

01:26 Positive would be the top, the negative would be the bottom.

01:30 So that's what the graph of that would look like if we were to graph both the positive and the negative.

01:35 Now what i'm going to do here in the next part is i'm going to go through and i'm going to take the derivative of each of those.

01:40 So i'm going to go through and i'm going to write the positive function, y equals positive one -third of the square root of nine plus x squared.

01:49 I'm going to take the derivative of that.

01:51 And then i'm also going to think about taking the derivative of the negative of that, negative 1 3rd square root of 9 plus x squared...

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