Derivatives calculus

Question

Find the First derivative of the following functions from the First Principles:



√4x+3

Answer 1

Is it (4x+3)^1/2 or (4x)^1/2+3 ?

Because I am not certain, I will instead show you how to do y=x^1/2 -- the procedure is similar either way:

f(x) = x^1/2

f'(x) = lim _h-->0 { ( (x+h)^1/2 - x^1/2 ) / h }

Multiply top and bottom by 1, where 1 = ( (x+h)^1/2+x^1/2 ) / ((x+h)^1/2+x^1/2 ):

= lim _h-->0 { ( ((x+h)^1/2 - x^1/2)((x+h)^1/2 + x^1/2 )) / ( ((x+h)^1/2+x^1/2 )•h ) }

Simplify:

= lim _h-->0 { ((x+h) + x^1/2(x+h)^1/2 - x^1/2(x+h)^1/2 - x ) / ( ((x+h)^1/2+x^1/2 )•h ) }

The numerator simplifies to just h:

= lim _h-->0 { h / ( ((x+h)^1/2+x^1/2 )•h ) }

The h in numerator and denominator cancel:

= lim _h-->0 { 1 / ( ((x+h)^1/2+x^1/2 ) ) }

Now just let h-->0:

= 1 / ( 2x^1/2 ) (ans)

Using f'(xⁿ) = nx^n-1 for f(x) = x^1/2 we get f'(x) = (1/2)x^-1/2 = 1/(2x^1/2) which matches, as expected.