1.

Find the derivative of the following functions 'ab initio', that is, using the definition. x^4

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SOLUTION :Let `y=x^4`
Then `dy/dx=lim_(deltaxto0)((x+DELTAX)^4-x^4)/(deltax)`
`x^4+4X^3 cdot deltax+6x^2 cdot(deltax)^2`
`=lim_(deltaxto0)(+4x(deltax)^3+(deltax)^4-x^4)/(deltax)`
`=lim_(deltaxto0){4x^3+6x^2 cdot deltax+4x(deltax)^2+(deltax)^3}=4x^3`


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