how to use the first principle to determine derivative of the function f(x)=square root(7x+5)

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We'll express the first principle of finding the derivative of a given function:

lim [f(x+h) - f(x)]/h, for h->0

We'll apply the principle to the given polynomial:

lim {sqrt [7(x+h)+5] - sqrt(7x+5)}/h

We'll remove the brackets from radicand:

lim [sqrt (7x+7h+5) - sqrt(7x+5)]/h

We'll multiply both, numerator and denominator, by the conjugate of numerator:

lim [sqrt (7x+7h+5) - sqrt(7x+5)][sqrt (7x+7h+5)+sqrt(7x+5)]/h*[sqrt (7x+7h+5)+sqrt(7x+5)]

We'll substitute the numerator by the difference of squares:

lim [(7x+7h+5) - (7x+5)]/h*[sqrt (7x+7h+5)+sqrt(7x+5)]

We'll eliminate like terms form numerator:

lim 7h/h*[sqrt (7x+7h+5)+sqrt(7x+5)]

We'll simplify and we'll get:

lim 7/[sqrt (7x+7h+5)+sqrt(7x+5)]

We'll substitute h by 0:

lim 7/[sqrt (7x+7h+5)+sqrt(7x+5)] = 7/[sqrt(7x+5)+sqrt(7x+5)]

We'll combine like terms from denominator:

**f'(x)=7/2sqrt(7x+5)**

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