`f(x) = sqrt(x + 4)` Find the derivative of the function by the limit process.

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Chapter 2, 2.1 - Problem 23 - Calculus of a Single Variable (10th Edition, Ron Larson).
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hkj1385 | (Level 1) Assistant Educator

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By limit process, the derivative of a function f(x) is :-

f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]

Now, the given function is :-

f(x) = sqrt(x+4)

THus, f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]

or, f'(x) = lim h ---> 0 [{(sqrt(x + h+ 4)) - sqrt(x+4)}/h] 

rationalizing the numerator we get

f'(x) = lim h ---> 0

[{(sqrt(x + h+ 4)) - sqrt(x+4)}*{(sqrt(x + h+ 4)) + sqrt(x+4)}/{h*{(sqrt(x + h+ 4)) + sqrt(x+4)}}] 

or, f'(x) = lim h ---> 0 [{(x+h+4) - (x+4)}/{h*{(sqrt(x + h+ 4)) + sqrt(x+4)}}] 

or, f'(x) = lim h ---> 0 [h/{h*{(sqrt(x + h+ 4)) + sqrt(x+4)}}]

or, f'(x) = lim h ---> 0 [1/{1{(sqrt(x + h+ 4)) + sqrt(x+4)}}]

putting the value of h = 0 in the above expression we get;

f'(x) = 1/{2*sqrt(x+4)}

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