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(t1 + δ t) − t1
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
general instant t on the time axis.
( dt )t L IMIT y(t + δ t) − y(t)
= δt→ 0 δt = δt→ 0 (t + δ t) − t
With the concept now clear, we define the INSTANTANEOUS RATE OF CHANGE of
some general function f(x) at some par ticular instant x1 .
par ( df )x
dx 1 L IMIT f(x1 + δ x) − f(x1)
= x → x δx = δx→ 0 δx = δx→ 0 (x + δ x) − x
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
general instant x on the x-axis.
LIMIT f(x + δ x) − f(x)
= δx→ 0 (x + δ x) − x df
= δx→ 0 δx
dx Here too we have to take the LIMIT both from the left and from the right . Even with
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
DIFFERENTIABLE at that instant.
With the definition of the FIRST DERIVATIVE in mind we proceed to differentiate the
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
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.
Also, we will be able to differ entiate WELL-BEHAVED functions. Finding the
DERIVATIVE by directly applying the definition is known as differentiation from
dif entiation from
f ir st principles .
It is not necessary to always differ entiate from FIRST PRINCIPLES. We may
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- Fall '09