Subsection1033 Summary There are four secondorder partial derivatives of a function f of two independent variables x and y fxx = (fx)x, fxy = (fx)y, fyx = (fy)x, and fyy = (fy)y The unmixed secondorder partial derivatives, fxx and fyy, tell us about the concavity of the traces
X/(x^2 y^2) partial derivative-Generalizing the second derivative Consider a function with a twodimensional input, such as Its partial derivatives and take in that same twodimensional input Therefore, we could also take the partial derivatives of the partial derivatives These are called second partial derivatives, and the notation is analogous to the notation forWith respect to threedimensional graphs, you can picture the partial derivative by slicing the graph of with a plane representing a constant value and measuring the slope of the resulting curve along the cut Intersecting y=0 plane with the graph
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{eq}f(x,y)=\frac{1}{\sqrt{x^{2}y^{2}}} {/eq} Partial Derivative The derivative of the composition of two or more variables with respect to one taking other as the constant is known as the partialLet's first think about a function of one variable (x) f(x) = x 2 We can find its derivative using the Power Rule f'(x) = 2x But what about a function of two variables (x and y) f(x, y) = x 2 y 3 We can find its partial derivative with respect to x when we treat y as a constant (imagine y is a number like 7 or something) f' x = 2x 0 = 2x
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