Leibniz ( - ) and - - Differentials and Partial Derivatives For instance, if we have two functions, say f x g x and suppose that we want to evaluate these functions at say x = . Which one will be easy to evaluate? Obviously, g ( . will be easier to calculate than f ( .
. If we are ready to accept some error in calculating f ( . , then we can find a linear function that approximates f near x = and use this linear function to obtain an approximate value of f ( . .We know that the graph of a function is a nonvertical line if and only if it is a linear function.
Out of infinitely many straight lines passing through any given point on the graph of the function, only tangent line gives a good approximation to the function, because the graph of f looks approximately a straight line on the vicinity of the point ( , ) . Fig. . Fig.
. Tangent Line From the figures above it is clear that among these straight lines, only the tangent line to the graph of f x ( ) at x = gives a good approximation near the point x = . Basically we are “linearizing” the given function at a selected point ( , ) . This idea helps us in estimating the change in the function value near the chosen point through the change in the input.
We shall use “derivative” to introduce the concept of “differential” which approximates the change in the function and will also be useful in calculating approximate values of a function near a chosen point. The derivative measures the instantaneous rate of change where as the differential approximates the change in the function values. Also, differentials are useful later in solving differential equations and evaluating definite integrals by the substitution method. After learning differentials, we will focus on real valued