find derivative using limit definition:
f(x) = x - sqrt(x)
so f'(x) =
lim
h->0 [f(x+h) - f(x)]/h
but I keep trying to solve by multiplying by the conjugate but I can't figure it out..there's nothing that can be cancelled or anything and I can't get the derivative
sorry this is a repost, but i messed up my other one and i really need help on this
The key to this is the binomial theorem:
(x+h)^n = x^n + n*x^(n-1)*h + n(n-1)/2 * x^(n-2) * h^2 + ...
So, we find ourselves with
(h + sqrt(x) - sqrt(x+h))/h
= (h + x^(1/2) - (x^(1/2) + 1/2 * x^(-1/2)*h + <higher powers of h>)/h
= 1 - 1/2 * x^(-1/2)
(all terms with h go to zero)
No problem, I'm here to help you out!
To find the derivative of the function f(x) = x - sqrt(x) using the limit definition, let's start by applying the difference quotient formula:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h.
Now let's plug in the function f(x) = x - sqrt(x) into this formula:
f'(x) = lim(h->0) [(x + h) - sqrt(x + h) - (x - sqrt(x))] / h.
Now simplify the expression under the limit by expanding and canceling out terms:
f'(x) = lim(h->0) [x + h - sqrt(x + h) - x + sqrt(x)] / h.
In this expression, you can see that the x terms cancel each other out:
f'(x) = lim(h->0) [h - sqrt(x + h) + sqrt(x)] / h.
Now, let's focus on the remaining term, -sqrt(x + h) + sqrt(x). We can multiply it by the conjugate to simplify further.
Multiply the numerator and denominator by the conjugate of the numerator, which is -sqrt(x + h) - sqrt(x):
f'(x) = lim(h->0) [(h - sqrt(x + h) + sqrt(x)) * (-sqrt(x + h) - sqrt(x))] / [h * (-sqrt(x + h) - sqrt(x))].
Expanding the numerator, we get:
f'(x) = lim(h->0) [-h * sqrt(x + h) + sqrt(x + h)^2 - sqrt(x) * (-sqrt(x + h)) - sqrt(x) * (-sqrt(x))] / [h * (-sqrt(x + h) - sqrt(x))].
Simplifying further:
f'(x) = lim(h->0) [-h * sqrt(x + h) + (x + h) - sqrt(x) * (-sqrt(x + h)) + sqrt(x)^2] / [h * (-sqrt(x + h) - sqrt(x))].
Now, in the limit as h approaches 0, we can simplify:
f'(x) = lim(h->0) [x + h - sqrt(x) * (-sqrt(x))] / [h * (-sqrt(x) - sqrt(x))].
f'(x) = lim(h->0) [x + h + x] / [h * (-2sqrt(x))].
f'(x) = lim(h->0) [2x + h] / [h * (-2sqrt(x))].
Now, as h approaches 0, we can cancel out the h terms:
f'(x) = lim(h->0) (2x + h) / (h * (-2sqrt(x))).
f'(x) = (2x) / (-2sqrt(x)).
Finally:
f'(x) = -sqrt(x).
So, the derivative of f(x) = x - sqrt(x) is -sqrt(x).