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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# The opera oper units measure operation with a

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Unformatted text preview: 0 (t1 + δ t) − t1 2 1 1 By dropping the subscript of t1 , we extend this concept of the INSTANTANEOUS RATE OF CHANGE of the par ticular function y(t) at par ticular instant t1 to any par par general instant t on the time axis. dy ( dt )t L IMIT y(t + δ t) − y(t) dy LIMIT δy = δt→ 0 δt = δt→ 0 (t + δ t) − t dt With the concept now clear, we define the INSTANTANEOUS RATE OF CHANGE of particular eneral some general function f(x) at some par ticular instant x1 . par ( df )x dx 1 L IMIT f(x1 + δ x) − f(x1) LIMIT δf LIMIT δf = x → x δx = δx→ 0 δx = δx→ 0 (x + δ x) − x 1 1 2 1 Again, dropping the subscript of x1 , we extend this concept of the INSTANTANEOUS RATE OF CHANGE of the general function f(x) at par ticular instant x1 to any particular eneral par general instant x on the x-axis. LIMIT f(x + δ x) − f(x) = δx→ 0 (x + δ x) − x df LIMIT δf = δx→ 0 δx dx Here too we have to take the LIMIT both from the left and from the right . Even with left right SINGLE VALUED and CONTINUOUS functions, it is quite possible when taking the limits to end up with something undefined such as +∞, −∞, division by zero and ∞/∞ , or the left LIMIT ≠ right LIMIT. In this case we say the function f(x) is not left right DIFFERENTIABLE at that instant. 55 differentia With the definition of the FIRST DERIVATIVE in mind we proceed to differentiate the dif entiate n n--1 2 general polynomial f(x) = a n x + a n-- 1 x + . . . + a 2 x + a 1 x + a 0 . In fact we need to know only how to differentiate x n , the general variable term differentia dif entiate of the general polynomial. We shall do this two ways : the Vedic way and the Western way. Once we know how to differentiate x n , we may differentiate any polynomial. differentia dif entiate Also, we will be able to differ entiate WELL-BEHAVED functions. Finding the differentia dif entiate DERIVATIVE by directly applying the definition is known as differentiation from differentia dif entiation from irst f ir st principles . It is not necessary to always differ entiate from FIRST PRINCIPLES. We may differentia dif entiate...
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