Module lecture calculus, Lecture notes of Calculus for Engineers

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MODULE 3

The Derivative; Symbols

The Differentiation Formulas

Differentiation of Algebraic Functions

A FUNCTION is said to be differentiable at x = xo if it has a derivative there. A function is said to

be differentiable on an interval if it is differentiable at every point of the interval.

The functions of elementary calculus are differentiable, except possibly at certain isolated points, on

their intervals of definition.

DIFFERENTIATION FORMULAS. In these formulas u, v and w are differentiable functions of x.

INVERSE FUNCTIONS. Let y=f(x) be differentiable on the interval a ≤ x ≤ b and suppose that d y d x does not change sign on the interval. Then from figure a and figure b the function assumes once and only once every value between f (a) = c and f (b) = d. Thus, for each value of y on the respective interval, there corresponds one and only one value of x and x is a function of y, say x = g(y). The functions y = f(x) and x = g(y) are called inverse functions

Differentiation of a function of a function

If y = f(u) and u= g(x), then y I fg(x)) is a function of x. If y is a differentiable function of u and if u is a

differentiable function of x, then y = f{g(x)} is a differentiable function of x and the derivative dy/dx

may be obtained by one of the following procedures:

Implicit Differentiation