How do I use the limit definition of derivative to calculate the following?

I will start with a simpler problem.

g(x) = sqrt(x). Let's see if we can show that

g'(x) = 1/[2*sqrt(x)]. As follows:

g'(x) = lim h->0 of

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

= lim h->0 of

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

{ h*[sqrt(x+h) + sqrt(x)] }

= lim h->0 of

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

= lim h->0 of

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

= lim h->0 of

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

= obviously 1/[2*sqrt(x)].

Next, I use the "quotient rule".

If the derivative of sqrt(x) is 1/[2*sqrt(x)],

then the derivative of 1/sqrt(x) is

{sqrt(x)*0 - 1*1/[2*sqrt(x)]} / x

= -1/[2*x^(3/2)].

Finally, if the derivative of 1/sqrt(x) is

-1/[2*x^(3/2)], then the derivative of 1/sqrt(3-x)

is -1/[2*(3-x)^(3/2)]*[d(3-x)/dx]

= 1/[2*(3-x)^(3/2)].

This is the "chain rule."

Use this trick

f(x) = 1/(sqrt(3-x))

f(x+h) = 1/sqrt(3-(x+h))

h is a very small quantity. Take out (3-x)

sqrt( 3-x) (1-h/(3-x)= sqrt (3-x) sqrt( 1-h/(3-x))

This you can write

sqrt(3-x) sqrt(1-h/2 * 1/(3-x)

= sqrt(3-x) (1-1/2 (h/(2(3-x)))

You have

f(x+h) = f(x) ( 1+h/(4 (3-x)). Here 1/(1-a) = 1+a when a is a very small quantity.

So the derivative

(f(x+h) -f(x))/h = f(x) 1/(4(3-x)) Try it.

At x=3, the function is interderminate and At x >3 , the function is complex

It's not that impossible. You just have to take it step-by-step.

f(x) = 1 / sqrt(3 - x)

f(x + h) = 1 / sqrt(3 - (x + h))

Let's just focus on f(x + h) - f(x) for now

1 / sqrt(3 - (x + h)) - 1 / sqrt(3 - x) =>

1 * sqrt(3 - x) / (sqrt(3 - x) * sqrt(3 - (x + h))) - 1 * sqrt(3 - (x + h)) / (sqrt(3 - x) * sqrt(3 - (x + h))) =>

(sqrt(3 - x) - sqrt(3 - (x + h))) / (sqrt(3 - x) * sqrt(3 - (x + h)))

Now normally, we rationalize the denominator, but for this case, we're going to rationalize the numerator.

a = sqrt(3 - x)

b = sqrt(3 - (x + h))

(a - b) / (a * b) =>

(a - b) * (a + b) / (a * b * (a + b)) =>

(a^2 - b^2) / (a * b * (a + b)) =>

((3 - x) - (3 - (x + h))) / (sqrt(3 - x) * sqrt(3 - x - h) * (sqrt(3 - x) + sqrt(3 - x - h))) =>

(3 - x - 3 + x + h) / (sqrt(3 - x) * sqrt(3 - x - h) * (sqrt(3 - x) + sqrt(3 - x - h))) =>

h / (sqrt(3 - x) * sqrt(3 - x - h) * (sqrt(3 - x) + sqrt(3 - x - h)))

Now divide by h. We have f(x + h) - f(x), so we need to divide by h

1 / (sqrt(3 - x) * sqrt(3 - x - h) * (sqrt(3 - x) + sqrt(3 - x - h)))

Let h go to 0

1 / (sqrt(3 - x) * sqrt(3 - x) * (sqrt(3 - x) + sqrt(3 - x))) =>

1 / ((3 - x) * 2 * sqrt(3 - x)) =>

1 / (2 * (3 - x)^(3/2))

See? It's not that bad.