17. Partial Derivatives
17.1 Increments
17.2 Partial Derivative dy
17.3 Partial Derivative, df (f not in terms of x)
17.4 Other Partial Derivatives
17.5 Higher Order Partial Derivatives
17.6 Total Derivatives
17.8 Finding Dy/Dx
17.2 Partial Derivative dy
17.3 Partial Derivative, df (f not in terms of x)
17.4 Other Partial Derivatives
17.5 Higher Order Partial Derivatives
17.6 Total Derivatives
17.8 Finding Dy/Dx
Another way to define a derivative is through increments. Recall that the derivative of a function f(x) is defined as:
This limit can be interpreted as the slope of a line tangent to the curve f(x), an object's velocity, or the instantaneous rate of change of an object that moves along the curve f(x) from one point to another. Using the notion of an increment, the numerator can be interpreted as the increment in y ( Δy ) while the denominator can be interpreted as the increment in x ( Δx ).
Therefore the limit of the difference quotient can be translated into
Therefore the limit of the difference quotient can be translated into
The smaller our increments of x are, the closer our approximate of the limit of f(x). Then we can say that
If we multiply both sides by Δx, we can define Δy (the differential in y) as:
where Δx is the differential in x.
The above equation implies that the differential dy (or differential of the function f, df) is a function consisting of two independent variables, x and dx. The change in x, dx, or both can affect the value of dy. For functions with more than one variable, the differential dy is best referred to as the partial derivative of y.
