Find the derivative of the function using the definition of derivative?

Question

You are given the following function.

f (x) = x + sqrt(x)

Find the derivative of the function using the definition of derivative.

f '(x) = ?????

please help

Answer 1

It is kind of difficult to type a math equation, but I guess I will try my best...

By definition, derivative is:

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

For this question,

f(x+h) = (x+h) + sqrt (x+h)
f(x) = x + sqrt(x)

Put it into the equation, you will get:

[(x + h) + sqrt (x+h) - (x + sqrt (x))] / h
= [h + sqrt (x + h) - sqrt (x)] / h
= 1 + [sqrt (x+h) - sqrt(x)] / h

Multiply the sqrt part by the fraction [sqrt (x+h) + sqrt(x)] / [sqrt (x+h) + sqrt(x)] (which is equal to 1), you will get:

= 1 + (h+x-x) / {h*[sqrt(x+h) + sqrt(x)]}

= 1+ h / {h*[sqrt(x+h) + sqrt(x)]}

Cancel out the h

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

Now, take the limit as h approaches 0

= 1 + 1/[sqrt (x+0) + sqrt(x)]

= 1 + 1 / (2*sqrt(x)) <== Which it the same answer you will get from taking the derivative using the other rules.

Answer 2

this can be solved by the first principle method of derivaties. all u have to do is put x+dx in the functionin place of x .& then subtract it from the real function & divide by dx.

Answer 3

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