Formula to find derivative of a function f(x) by first principle :
This is also called as limit definition of the derivative.
Derivative of ^{3}√x using first principle :
Let
f(x) = ^{3}√x
To get rid of the cube root in ^{3}√(x + h) - ^{3}√x, we have to use the following algebraic identity.
a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})
If a = ^{3}√(x + h) and b = ^{3}√x, we have
Multiply both numerator and denominator by
Use the following algebraic identity in numerator part.
a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})
Therefore,
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