 Hello, and welcome to this screencast on section 10.3, second-order partial derivatives. As we saw last section, a function f of two independent variables x and y has two first-order partial derivatives, the first-order partial derivative with respect to x, and the first-order partial derivative with respect to y. We can again take partial derivatives of these two first-order partial derivatives to give us a total of four second-order partial derivatives. Taking the partial derivative with respect to x twice gives us this second-order partial derivative. The equal signs here are showing the four different notations you might see for this second-order partial derivative. Taking the partial derivative with respect to y twice gives us this second-order partial derivative. Taking the partial derivative with respect to x first then y gives us this second order partial derivative and lastly taking the partial derivative with respect to y first then x gives us this second order partial derivative. The first two second order partial derivatives are called unmixed while the second two are called mixed referring to that they contain a mix of x and y partial derivatives. Recall from single variable calculus that the second derivative measures the instantaneous rate of change of the derivative. This observation is key to understanding the meaning of the second order partial derivative. Consider this second order partial derivative with respect to x. Since this derivative requires us to hold y constant and differentiate twice with respect to x we may view this derivative as the second derivative of the trace of f where y is fixed. So this second order partial will measure the concavity of this trace. As an example consider this function f. The figure shows the graph of this function along with the trace when y is held constant at negative 1.5. Also shown are three tangent lines to this trace in blue with increasing x values from left to right among the three plots. As you can see from left to right the slopes of the tangent lines are decreasing as x increases. This tells us that the first order partial derivative with respect to x is decreasing which in turn tells us that the unmixed second order partial derivative with respect to x is negative. These observations go along with the fact that we can see that the trace is concave down. We can further verify that the unmixed second order partial derivative with respect to x along this trace is negative for all x values pictured by examining the function. In your work in section 10-3 you'll investigate the behavior of the remaining second order partial derivatives.