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Differentiation by Abinitio


Differentiation by Abinitio

This is also known as Differentiation by definition or Differentiation by Principle.

The fundamental method to find derivatives of a function y=f(x) with respect to x is called the First principle to find derivatives of a function.

Consider a function y= f(x)

Now if x is incremented by a value Δ x then the value of y also changes (say) by Δ y.

y + Δy = f(x+Δx)

Thus the change in value of y, is

Δy = (y + Δy) -y

?y/?x=(f(x+?x)-f(x))/?x

The derivative of function y = f(x) at a point (x, y) is the slope of the tangent of the function at that point.

dy/dx = lim?x→0 ?y/?x = lim?x→0 (f(x+?x)-1(x))/?x

Hence the derivative of the function y=f(x) is found by the above method.

Illustration:

Find the derivatives of the function y = x2 from first principle.

It is given that y=x2

If x is incremented by Δx then y changes by Δy

y + Δy = (x+Δx)2

The changes in y that is, Δy can be calculated as

 Δy = (y+Δy)-y=(x+Δ)2=2x ?x+(?x)2

Δy/?x = (2x?x+(?x)2)/?x 2x + ?x

dy/dx = lim?x→0 ?y/?x = lim?x→0 (2x+?x) = 2x

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