Speaker A
We're given the function f of x, y, and asked to find the partial derivatives and evaluate at the given point.
Learn how to find first and second order partial derivatives of a function of two variables and interpret their geometric meanings.
Key Takeaways
- First order partial derivatives represent slopes of tangent lines in the x and y directions.
- Second order partial derivatives indicate concavity of the surface in respective directions.
- Mixed partial derivatives describe how the slope in one direction changes as we move in the other direction.
- Mixed partial derivatives are equal if they are continuous over an open region.
- Evaluating derivatives at a point provides local geometric information about the surface.
What the video covers
- Introduction to finding first and second order partial derivatives of a function f(x,y).
- Explanation of first order partial derivatives as slopes of tangent lines in x and y directions.
- Step-by-step differentiation treating one variable as constant to find partial derivatives with respect to x and y.
- Evaluation of partial derivatives at a specific point (x=1/2, y=1).
- Graphical interpretation of tangent lines and slopes on the surface.
- Explanation of second order partial derivatives and their relation to concavity in x and y directions.
- Definition and interpretation of mixed partial derivatives and their equality under continuity conditions.
- Calculation and evaluation of second order partial derivatives and mixed partials at the given point.
- Geometric verification of concavity and slope changes using graph and tangent planes.
- Summary of how these derivatives describe the behavior of the surface locally.
Chapters
- 00:00Introduction and problem setup
- 00:13Understanding first order partial derivatives
- 00:53Finding partial derivative with respect to x
- 01:50Evaluating partial derivative with respect to x at the point
- 02:19Graphical interpretation of partial derivative in x direction
- 03:02Finding partial derivative with respect to y
- 03:41Evaluating partial derivative with respect to y at the point
- 04:17Graphical interpretation of partial derivative in y direction
- 04:43Introduction to second order partial derivatives
- 05:12Finding second order partial derivatives and mixed partials
Full Transcript — Download SRT & Markdown
Speaker A
Notice how we have two first order partial derivatives and two second order partial derivatives.
Speaker A
Let's begin by reviewing what the first order partial derivatives tell us about our surface, as well as how to find them.
Speaker A
The first order partial derivative of a function of two variables tells us the slope of the tangent line at a given point in either the positive x or positive y direction.
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This measures the rate of change of z, or a function f, with respect to x, or the rate of change of z, or f, with respect to y, based upon which partial derivative we're determining.
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To determine the partial derivative with respect to x, we consider y a constant and differentiate with respect to x.
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To determine the partial derivative with respect to y, we consider x a constant and differentiate with respect to y.
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So, going back to our example, we'll begin by determining a partial derivative with respect to x.
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So, we'll differentiate each term with respect to x, treating y as a constant. So, the derivative of five x squared with respect to x would be 10x, plus the derivative of three x squared y cubed with respect to x,
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treating y cubed as a constant. So, we'd multiply by two, and then subtract one from the exponent on x, which would give us six x to the first, or six x, y to the third.
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And then the derivative of three y to the fifth with respect to x would be zero.
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Again, because you're treating y as a constant. So, here is our first order partial derivative with respect to x, and now we'll evaluate this at our point.
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Notice how we only need the x and y values. So, we'll substitute 1/2 for x, and one for y.
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So, we'd have 10 times 1/2, plus six times 1/2, times one cubed, which would give us five, plus three, which equals eight, which means the slope of the tangent line in the x direction at this point is positive eight.
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And let's look at this graphically. Our surface is graphed here in purple. The point we're considering is this red point, and this blue line is the tangent line at that point in the x direction, which is in this yellow plane,
Speaker A
which is parallel to the x-axis. This arrow here indicates the positive x direction. Notice, as we move in this direction from our point, the line is moving uphill, and it has a slope of positive eight.
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So, this is what the partial derivative with respect to x is telling us about the tangent line at that point.
Speaker A
Now, we want to find the partial derivative with respect to y. So, we'll differentiate with respect to y, treating x as a constant.
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So, the derivative of five x squared with respect to y would be zero. The derivative of three x squared y cubed with respect to y, we would multiply by three, so that would give us nine.
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Then we'd have x squared y to the second because we subtract one from the exponent of three on the y.
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And then plus the derivative of three y to the fifth with respect to y would be 15 y to the fourth.
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And now, we'll evaluate this at our point. So again, we'll substitute 1/2 for x, and one for y.
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So, we'll have nine times 1/2 squared, times one squared, plus 15 times one to the fourth.
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So here, we'd have 9/4 plus 15, 15 over one, common denominator would be four.
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So, we have 9/4 plus 60/4, or 69/4. So, the slope of the tangent line in the positive y direction at this point would be positive 69/4.
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Let's take a look. Again, here is a graph of our surface. Now the plane is parallel to the y-axis, and this arrow here indicates the direction of the positive y-axis.
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So, notice how the tangent line is graphed here in blue. Notice if we move in the positive y direction, the slope of the tangent line would be positive 69/4.
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Notice how it's going uphill very quickly. Now, let's review what the second order partial derivatives tell us about a surface and how to find them.
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If the second order partial with respect to x is positive, then our function is concave up in the x direction.
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If the second order partial with respect to y is positive, then the function is concave up in the y direction.
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And of course, if these are negative, this would indicate that the function would be concave down in the x and y directions.
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The mixed partials, indicated using this notation here, tells us how a partial in one variable is changing in the direction of the other.
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So, the partial derivative of f with respect to x, and then with respect to y, tells us how fast the rate of change of f in the x direction is changing as we move in the y direction.
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We could also say this tells us how the slope of the tangent line in the x direction is changing as we move in the y direction.
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And the mixed partials are equal if both mixed partials are continuous on an open region for every point in that region.
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So, to find the second order partial with respect to x, we want to differentiate with respect to x twice.
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So, we'll take our first order partial with respect to x here, and differentiate with respect to x again, treating y as a constant.
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So, the derivative of 10x with respect to x would be 10, plus the derivative of six x, y to the third with respect to x, so we're treating y to the third as a constant.
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So, our derivative with respect to x would just be six y to the third.
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And now, we'll evaluate this at our point. So, we'd have 10 plus six times one to the third, which equals 16.
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So, notice how the second order partial with respect to x is positive, or greater than zero, meaning from this point, if we move in the x direction, the function is concave up.
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And let's go ahead and verify that. Notice from this view, where the yellow plane is parallel to the x-axis, we can see that the surface is concave up in the x direction.
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And then finally, we want to find the partial derivative with respect to x, then with respect to y, which means now we'll differentiate this first order partial with respect to x with respect to y.
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So, we'll treat x as a constant. So, the derivative of 10x with respect to y would be zero.
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The derivative of six x, y to the third with respect to y would be 18 x y squared.
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So, let's evaluate this at our point. So, we'd have 18 times 1/2, times one squared.
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So, this would be nine. And again, notice how this is positive, or greater than zero, which means the slope of the tangent line in the x direction, as we move in the y direction, is increasing.
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And before we go, let's go ahead and find the other second order mixed partial.
Speaker A
The partial derivative of f with respect to y, then with respect to x, which means we would now find the derivative of this partial derivative with respect to x.
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So, we treat y as a constant. So, the derivative of nine x squared y squared with respect to x would be 18 x to the first y squared, or 18 x y squared.
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And the derivative of 15 y to the fourth with respect to x would be zero.
Speaker A
So, notice how our second order and mixed partials are equal. I hope you found this helpful.
Topics:partial derivativesfirst order partial derivativesecond order partial derivativemixed partial derivativesconcavitytangent line slopemultivariable calculusfunction of two variablesMathispower4ucalculus tutorial
