Calculus Question..Find derivative using the definition

G(d) = 1/sqrt(d)

I got 1/2(sqrt(d))^2 but I guess that's not the right answer

can someone help me by doing this step by step please

The definition is lim(h->0) [G(d+h) - G(d)]/h

G(d+h) = 1/sqrt(d+h)
G(d) = 1/sqrt(d)

G(d+h) - G(d) = 1/sqrt(d+h) - 1/sqrt(d) = [sqrt(d) - sqrt(d+h)]/[sqrt(d+h) * sqrt(d)]

I got that just by putting them over a common denominator.

Now, how the heck can you simplify that? Answer: By multiplying numerator and denominator by [sqrt(d) + sqrt(d+h)]. Then you'll have [sqrt(d)^2 - sqrt(d+h)^2] = d - (d + h) in the numerator and the d's cancel out, leaving you -h. When you divide this by h, that will leave you with -1.

In the denominator you'll have sqrt(d+h) * sqrt(d) * [sqrt(d+h) + sqrt(d)]. In the limit as h->0, all the sqrt(d+h) terms become sqrt(d).

1/sqrt(d) is the same as d^(-1/2).
G(d) = d^(-1/2)
G'(d) = -1/2 (d)^(-3/2) [you multiply by the exponent (-1/2) and subtract 1 from the exponent so the previous (-1/2) goes to (-3/2).] So...
G'(d) = -1/(2(d)^3/2).

Simply taking the derivative you would get:

G ' (d) = -1/2 d^(-3/2) note G(d)=d^(-1/2) and you differentiate from there.

Using the definition is a bit more complicated:

= -1/2(sqrt(d))^-1.5


maybe!!!