How do you find f'(x) using the definition of a derivative for f(x)=sqrt(1+2x)f(x)=√1+2x?

1 Answer
George C.
Oct 8, 2015

Use definition:

f'(a) = lim_(h->0) (f(a+h) - f(a))/h

to find: f'(x) = 1/sqrt(1+2x)

Explanation:

Let f(x) = sqrt(1+2x)

Then the derivative at x=a is defined as the following limit:

f'(a) = lim_(h->0) (f(a+h) - f(a))/h

= lim_(h->0) (sqrt(1+2(a+h)) - sqrt(1+2a))/h

= lim_(h->0) (sqrt(1+2(a+h)) - sqrt(1+2a))/h * (sqrt(1+2(a+h)) + sqrt(1+2a))/(sqrt(1+2(a+h)) + sqrt(1+2a))

= lim_(h->0) (sqrt(1+2(a+h)) - sqrt(1+2a))/h * (sqrt(1+2(a+h)) + sqrt(1+2a))/(sqrt(1+2(a+h)) + sqrt(1+2a))

= lim_(h->0) ((1+2(a+h)) - (1+2a))/(h(sqrt(1+2(a+h)) + sqrt(1+2a)))

= lim_(h->0) (2color(red)(cancel(color(black)(h))))/(color(red)(cancel(color(black)(h)))(sqrt(1+2(a+h)) + sqrt(1+2a)))

= lim_(h->0) 2/(sqrt(1+2(a+h)) + sqrt(1+2a))

= 2/(sqrt(1+2a) + sqrt(1+2a))

= 1/sqrt(1+2a)

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

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