 # Application of Derivatives

## Applications of derivatives are many in our real life. We apply derivatives to approximation, ‘rate of change’, increasing and decreasing functions, and slope of a tangent and a normal. Rate of Change: If a quantity y varies with another quantity x, satisfying some rule y = f(x), then dy/dx represents the rate of change of y with respect to x. • First Derivative Test For Maxima and Minima: Let f (x) be a function defined on an open interval I. Let f (x) be continuous at a critical point ‘a’ in I. Then (1) If f ′(x) changes sign from positive to negative as x increases through a, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of a, and f ′(x) < 0 at every point sufficiently close to and to the right of a, then a is a point of local maxima. (2) If f′(x) changes sign from negative to positive as x increases through a, i.e., if f′(x) < 0 at every point sufficiently close to and to the left of a, and f ′(x) > 0 at every point sufficiently close to and to the right of a, then a is a point of local minima. • Second Derivative Test For Maxima and Minima: Let f(x) be a function defined on an interval I and a Î I. Let f(x) be twice differentiable at a. Then (i) x = a is a point of local maxima if f ′(a) = 0 and f ″(a) < 0 The values f(a) is local maximum value of f (x). (ii) x = a is a point of local minima if f′(a) = 0 and f ″(a) > 0. In this case, f (a) is local minimum value of f (x). (iii) The test fails if f'(a) = 0 and f" (a) = 0. Keywords: Increasing function, decreasing function, slope of tangent, equation of tangent,

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