In each case, treat all variables as constants except the one whose partial derivative you are calculating. We can use a contour map to estimate partial derivatives of a function g ( x, y ). We now return to the idea of contour maps, which we introduced in Functions of Several Variables. If we wish to find the slope of a tangent line passing through the same point in any other direction, then we need what are called directional derivatives, which we discuss in Directional Derivatives and the Gradient. ![]() Therefore, ∂ f / ∂ x ∂ f / ∂ x represents the slope of the tangent line passing through the point ( x, y, f ( x, y ) ) ( x, y, f ( x, y ) ) parallel to the x -axis x -axis and ∂ f / ∂ y ∂ f / ∂ y represents the slope of the tangent line passing through the point ( x, y, f ( x, y ) ) ( x, y, f ( x, y ) ) parallel to the y -axis. If we choose to change y y instead of x x by the same incremental value h, h, then the secant line is parallel to the y -axis y -axis and so is the tangent line. As h h approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the slope of the secant line represents an average rate of change of the function f f as we travel parallel to the x -axis. This line is parallel to the x – z plane. This raises two questions right away: How do we adapt Leibniz notation for functions of two variables? Also, what is an interpretation of the derivative? The answer lies in partial derivatives.į ( x + h, y ) − f ( x, y ) h. For a function z = f ( x, y ) z = f ( x, y ) of two variables, x x and y y are the independent variables and z z is the dependent variable. Leibniz notation for the derivative is d y / d x, d y / d x, which implies that y y is the dependent variable and x x is the independent variable. When studying derivatives of functions of one variable, we found that one interpretation of the derivative is an instantaneous rate of change of y y as a function of x. Derivatives of a Function of Two Variables This carries over into differentiation as well. However, we have already seen that limits and continuity of multivariable functions have new issues and require new terminology and ideas to deal with them. Finding derivatives of functions of two variables is the key concept in this chapter, with as many applications in mathematics, science, and engineering as differentiation of single-variable functions. Now that we have examined limits and continuity of functions of two variables, we can proceed to study derivatives. 4.3.4 Explain the meaning of a partial differential equation and give an example.4.3.3 Determine the higher-order derivatives of a function of two variables.4.3.2 Calculate the partial derivatives of a function of more than two variables. ![]()
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