defenition of derivative?

Question

(2/x^2)-sqrt(4), what is this when you use the defenition of derivative. I cant get it simplified all the way down.

Answer 1

adjective
Resulting from or employing derivation: a derivative word; a derivative process.
Copied or adapted from others: a highly derivative prose style.
noun
Something derived.
Linguistics A word formed from another by derivation, such as electricity from electric.
Mathematics
The limiting value of the ratio of the change in a function to the corresponding change in its independent variable.
The instantaneous rate of change of a function with respect to its variable.
The slope of the tangent line to the graph of a function at a given point. Also called differential coefficient, fluxion
Chemistry A compound derived or obtained from another and containing essential elements of the parent substance.

Answer 2

Mathematically, the derivative of the function f with respect to the variable x is the function f' whose value at x is:

f'(x) = lim (f(x+h) - f(x)) / h as h approaches zero, provided the limit exits.

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

Therefore the derivative:
f'(x) = lim { [(2/(x + h)^2) - 2] - [(2/x^2) - 2] } / h
as h approaches zero.

Simplify:
f'(x) = lim { (2/(x + h)^2) - 2/x^2 } / h

To simplify the numerator, multiply the left term by x^2/x^2, and the right term by (x+h)^2/(x+h)^2, so that we can combine the terms in the numerator.

This results in:
f'(x) = lim {{[2x^2]/[x^2(x+h)^2]} - {[2(x+h)^2]/[x^2(x+h)^2]}} / h

Combine terms:
f'(x) = lim {[2x^2 - 2(x+h)^2] / [x^2(x+h)^2]} / h

Bring the h in the denominator into the upper fraction.
f'(x) = lim [2x^2 - 2(x+h)^2] / [hx^2(x+h)^2]

Expand -2(x+h)^2 and the 2x^2 terms cancel, leaving:
f'(x) = lim (-4xh - 2h^2) / hx^2(x+h)^2

Factor out h from numerator and denominator:
f'(x) = lim h(-4x - 2h) / h[x^2(x+h)^2]

Cancel h's.
f'(x) = lim (-4x - 2h) / x^2(x+h)^2

Expand denominator
f'(x) = lim (-4x - 2h) / x^2(x^2 + 2xh + h^2)

Let h approach zero:
f'(x) = -4x / x^4

x's cancel and leave the answer:
f'(x) = -4 / x^3

Hope this is what you were looking for! :)

Answer 3

Derivatives are simply rates of change. How something changes, increases decreases and how it increases and decreases. You can do many things with it. See Wikipedia for help. And what do you want to do with your function? Take the derivative? It would be (-4/x^4)
The sqrt(4) would simply be left out because it's still a constant. Constants have zero slope. That's basically what Derivatives are, instantaneous slope.

Answer 4

if you simplify that equation it is:

2/(x^2)-2

differentiate by parts (keep in mind that the derivative of a constant is 0)

for the first part, use the quotient rule. (bottom*deriv of the top - top*deriv of the bottom...all divided by the bottom squared)

you should get something like this:
[(x^2)(0)-(2)(2x)]/(x^4)
= -4x/(x^4)

factor out an x and you get:
-4/(x^3)

Answer 5

I am going to assume that you mean

2/(x^2) - 2, not

(2/x)^2 - 2

First of all, remember that (1/x)^n is the same as x^(-n).

Rewrite your equation:

2/(x^2) - 2 is the same as
2(x^(-2)) - 2

So the derivative is

-4/(x^3)

Its the same as the answer above but my way was much easier.

Answer 6

sqrt of 4 is 2 just to let u know.

Answer 7

google it