Derivative Calculus Question using the limit Theorem?

Question

The starting equation was 1/√x which i plugged into the theorem and got this {(x^(1/2) - (x+h)^(1/2)}/{h(x+h)^1/2(x)^(1/2)} So i changed it to ^1/2 to make it eaiser. But i can not figure out how to get the h out of the top to cancel out the bottom so i can plug 0 in for h. Any help on the 2nd step would be wonderful.

Thanks,

Brad

Answer 1

f(x) = 1/√x = x^(-1/2)

f'(x) = lim(h->0) [f(x+h) - f(x)] / h
= lim(h->0) [(x+h)^(-1/2) - x^(-1/2)] / h
= lim(h->0) [(x^(-1/2)) . (1+h/x)^(-1/2) - x^(-1/2)] / h
Now we expand the term (1+h/x)^(-1/2) as an infinite series:
(1+h/x)^(-1/2) = 1 + (-1/2) (h/x) + (-1/2)(-3/2) (h/x)^2/2! + ....
So f'(x) = lim(h->0) [(x^(-1/2)) . {1 + (-1/2) (h/x) + (-1/2)(-3/2) (h/x)^2/2! + ....} - x^(-1/2)] / h
= lim(h->0) [(x^(-1/2)) . {(-1/2) (h/x) + (-1/2)(-3/2) (h/x)^2/2! + ....}] / h
= lim(h->0) (x^(-1/2)) . {(-1/2) (1/x) + (-1/2)(-3/2) (h/x^2)/2! + ....}
= (x^(-1/2)) . {(-1/2) (1/x)}
(since all the subsequent terms have a power of h in them)
= (-1/2) x^(-3/2).

Answer 2

f(x) = 1/x^1/2

f(x+h) = 1/(x+h)^1/2

f(x+h) - f(x) = (1/(x+h)^1/2) - (1/x^1/2)

multiply RHS with (x)^1/2 *(x+h)^1/2

f(x+h) - f(x) = [x^1/2 - (x+h)^1/2)]/(x)^1/2(x+h)^1/2

multiply RHS with [(x^1/2 + (x+h)^1/2][(x)^1/2 + (x+h)^1/2

f(x+h) - f(x) = [x - x - h)]/(x)^1/2(x+h)^1/2 [(x)^1/2 + (x+h)^1/2]

f(x+h) - f(x) = -h/(x)^1/2(x+h)^1/2 [(x)^1/2 + (x+h)^1/2]


[f(x+h) - f(x)]/h = -1/((x)^1/2(x+h)^1/2 [(x)^1/2 + (x+h)^1/2]

h in the numerator and denominator gets cancelled out

when h approaches zero, the expression becomes

-1/(x)^1/2* (x)^1/2[(x)^1/2 + (x)^1/2]

=>-1/(x)(2x^1/2)

=> - 1/2 x^3/2

=> (-1/2) x^(-3/2)

Answer 3

Take a break... have a cuppa tea.. then put h over 1 and see how you go... sometimes just walking away from an equation helps to solve it.

Answer 4

f(x+h) - f(x)
------- ------- =
..... h

1/sqrt(x+h) - 1/sqrt(x)
----------- ---------------- =
.............h

This is where I start to get a tad lost... you may want to check w/ examples from your book...

Cross multiply and multiply the denominator by the product of the denominators:

sqrt(x) - sqrt(x+h)
----------- ---------------- =
h (sqrt(x)(sqrt(x+h))

sqrt(x) - sqrt(x+h)
----------- ---------------- =
h*sqrt[(x)(x+h)]

sqrt(x) - sqrt(x+h)
----------- ---------------- =
h*sqrt(x^2+xh)


-1
----------- ---------------- =
sqrt(x^2+x)

The answer should be...
-1
-------------- =
2 sqrt(x^3)