Derivatives find the derivative of y2x412 using the first pr

Derivatives

find the derivative of y=(2x+4)^1/2 using the first principle

Solution

First principle states:

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

We\'ll apply the principle to the given polynomial:

lim {sqrt [2(x+h)+4] - sqrt(2x+4)}/h

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

lim [sqrt (2x+2h+4) - sqrt(2x+4)]/h

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

lim [sqrt (2x+2h+4) - sqrt(2x+4)][sqrt (2x+2h+4)+sqrt(2x+4)]/h*[sqrt (2x+2h+4)+sqrt(2x+4)]

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

lim [(2x+2h+4) - (2x+4)]/h*[sqrt (2x+2h+4)+sqrt(2x+4)]

We\'ll eliminate like terms form numerator:

lim 2h/h*[sqrt (2x+2h+4)+sqrt(2x+4)]

We\'ll simplify and we\'ll get:

lim 2/[sqrt (2x+2h+4)+sqrt(2x+4)]

We\'ll substitute h by 0:

lim 2/[sqrt (2x+2h+4)+sqrt(2x+4)] = 2/[sqrt(2x+4)+sqrt(2x+4)]

We\'ll combine like terms from denominator:

f\'(x)=1/sqrt(2x+4)