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Question

Differentiate xex From First Principles.


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Solution

Step 1: Find the derivative of f(x)=xex

Given,xex

Let , f(x)=xex

Derivative of a function f(x) is given by:

f(x)=limh0f(x+h)f(x)h

{where h is a very small positive number}

The derivative of f(x)=xex is given as:

f(x)=limh0f(x+h)f(x)hf(x)=limh0(x+h)e(x+h)xexh

Now by using the algebra of limits:

f(x)=limh0hex+hh+limh0xex+hexhf(x)=limh0ex+h+limh0xexeh1h

Step 2: Again by using the algebra of limits:

f(x)=ex+0+limh0eh1h×limh0xex

Now by using the formula:

limx0ex1x=logee=1f(x)=ex+xexf(x)=ex(x+1)

Therefore, the derivation of f(x)=xex=ex(x+1).


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