Introduction
Suppose that ƒ is a function of more than one variable. For instance,
The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the xz-plane, and those that are parallel to the yz-plane (which result from holding either x or y constant, respectively.)
To find the slope of the line tangent to the function at (1, 1, 3) that is parallel to the xz-plane, the y variable is treated as constant. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point (x, y, z) is found to be:
So at (1, 1, 3), by substitution, the slope is 3. Therefore
at the point. (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.
Read more about this topic: Partial Derivative
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