Partial Derivatives

    In general, a derivative is a rate of change or a slope of a tangent line; and the process of differentiating a several-variable function with respect to one independent variable, by using the rules from the calculus of one variable, is called partial differentiation. This topic illustrates how partial derivatives can be interpreted geometrically as the slopes of the tangent lines at a given point to the traces of a surface in the planes determined by the coordinates of the given point. Higher-order partial differentiation of functions of several variables and Clairaut's Theorem are also detailed.

Definition (Partial Derivative of a Multivariate Function) If is a function of the variables ,  then the partial derivative of with respect to is the function defined by

.

    For the partial differentiation of a function of two variables, , we find the partial derivative with respect to by regarding as a constant while differentiating the function with respect to . Similarly, the partial derivative with respect to is found by regarding as a constant while differentiating with respect to .

Definition (Partial Derivative of a Two Variable Function) If , then the partial derivatives of with respect to and are the functions and , respectively, defined by


and


The partial derivatives and are denoted by


and

Example (Partial Derivative of a Two Variable Function) Find the following partial derivatives.

(a) Find and given .

    Solution. Holding constant and differentiating with respect to , we get and so Holding constant and differentiating with respect to , we get and so

(b) Find and given .

    Solution. We have and ; and so and The graph of is the paraboloid and the vertical plane intersects it in the parabola The slope of the tangent line to this parabola at the point is . Similarly the curve in which the plane intersects the paraboloid is the parabola and the slope of the tangent line at is .
    


(c) If , calculate and .

    Solution. Using the chain rule for functions of one variable, we have


and



(d) Find and if is defined implicitly as a function of and by the equation

    Solution. To find we differentiate implicitly with respect to , being careful to treat as a constant:



Solving this equation for we obtain



Similarly, implicit differentiation with respect to gives

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Partial Derivatives
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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