Page 6 - 12-Math-7 Vectors

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lim d y = lim f ( x + d x) - f ( x) (vi)
dd x→0 x d x→0 dx

lim d y is denoted by dy , so (vi) is written as dy = f '(x)
d x dx dx
d x→0

Note: The symbol dy is used for the derivative of y with respect to x and here it is not a
dx

quotient of dy and dx. dy is also denoted by y ’.
dx

Now we write, in a table the notations for the derivative of y = f ( x)used by different

mathematicians:

Name of Leibniz Newton Lagrange Cauchy

Mathematician

Notation used for derivative dy df f (x) f ' ( x) Df ( x)
dx or dx

If we replace x + d x by x and x by a, then the expression

f ( x + d x) - f ( x) becomes f ( x) - f (a) . and the change d x in the independent variable, in this

case, is x - a .

So the expression f (x +d x)- f (x) is written as f (x)- f (a) (vii)

dx x-a

Taking the limit of the expressiom(vii) when x → a , gives

lim f ( x) - f (a) = f ' (a). Here f ' (a)

x→a x - a
is called the derivative of f at x = a .

version: 1.1

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