Derivative of f(x) = 1/sqrt(x). I need to find this derivative using limits as opposed to just shortcuts.?

Question

The answer should be -(1/(2x^3/2)) but I'm having trouble getting to it using limits.

I have to use limits on this question which makes it very annoying :( Any help on where to start would be appreciated

Answer 1

derivative of f(x) is defined as

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

here f(x) = 1/√x

f(x + h) = 1/√(x+h)

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

make denominator common as √x (√x+h)

=>[√x - (√x+h)]/√x (√x+h)

multiply with [√x + (√x+h)]/[√x + (√x+h)]

[x - (x+h)]/√x (√x+h)[√x + (√x+h)]

=> -h/[x(√x+h + (x+h)√x

divide by h

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

when h-> 0, the expression becomes

f'(x) = -1/[x(√x+0) +(x+0)√x]

=>-1/2x√x

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

Answer 2

OK, fair enough, but this will be tricky to write in ASCII.

Derivative is:
lim(h->0) {[1/sqrt(x+h) - 1/sqrt(x)] / h}

I won't write the lim from now on, but remember it's there.

Let's find the common denominator for the numerator, so the expression above becomes:

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

I hope it's clear up to this moment.

Now we'll do a little trick: multiply this by

[sqrt(x) + sqrt(x+h)] / [sqrt(x) + sqrt(x+h)] ,

which is clearly 1, so it's OK.

Now the magic comes in, because if you remember

a^2 - b^2 = (a + b) * (a - b).

See how this helps?

If you make the calculation after multiplying by our "one", you will get that the derivative is:

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

Taking h to its limit (zero), you get:

- 1/{2[x * sqrt(x)]}

which is the answer you quoted.

Answer 3

I assume you're talking about the limit definition of a derivative.

The critical part is that you need to rationalize the numerator by multiplying top and bottom of the fraction (whose limit you seek) by the conjugate of the numerator.

Answer 4

f(x) = 1/√x

f'(x) = lim [1/√(x+h) - 1/√(x)] / h

= lim [√(x) - √(x+h)] / h √(x+h) √(x)
{rationalize the numerator...}
= lim [√(x) - √(x+h)][√(x) + √(x+h)] / h √(x+h) √(x) [√(x) + √(x+h)]

= lim -h / h √(x+h) √(x) [√(x) + √(x+h)]

= -1/ x [2√x]
= -1/ 2 x^(3/2)


§

Answer 5

[1/x^(.5+h) - 1/x^.5]/h
lim h--> 0
= (x^.5-x^(.5+h))/(h(x^.5x^(x+h))) = 0/0
lim h --> 0
Now you can use L'Hospital's Rule to find limit