# Evaluate derivative of `y=sqrt(3x+1)` using the first principle.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First, 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 first principle to the given polynomial:

lim {sqrt [3(x+h)+1] - sqrt(3x+1)}/h

The next step is to remove the brackets under the square root:

lim [sqrt (3x+3h+1) - sqrt(3x+1)]/h

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

lim [sqrt (3x+3h+1) - sqrt(3x+1)][sqrt (3x+3h+1)+sqrt(3x+3h+1)]/h*[sqrt (3x+3h+1)+sqrt(3x+1)]

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

lim [(3x+3h+1) - (3x+1)]/h*[sqrt (3x+3h+1)+sqrt(3x+1)]

We'll eliminate like terms form numerator:

lim 3h/h*[sqrt (3x+3h+1)+sqrt(3x+1)]

We'll simplify and we'll get:

lim 3/[sqrt (3x+3h+1)+sqrt(3x+1)]

We'll substitute h by 0:

lim 3/[sqrt (3x+3h+1)+sqrt(3x+1)] = 3/[sqrt(3x+1)+sqrt(3x+1)]

We'll combine like terms from denominator:

f'(x)=3/2sqrt(3x+1)