What is the meaning of derivative?
596305
I means that something derives or comes from something else.i.e. some English words are a derivative of Latin ones, therefore they come from certain Latin words.
874034
It is the first derivative.
546678
Hopefully you'll find this site helpful.\n\nhttp://www.sosmath.com/calculus/calculus.html\n\nThere is a lot of information and example problems on limits and derivatives.  Good luck with your calculus class!
763677
the derivative of a function is the slope of the tangent line at any given point or the rate of change at any given point.  the slope of the tangent line of any point on f(x) where x=a is equal to  lim x->a of (f(x)-f(a))/(x-a)
931778
Since you have conflicting answers, I'm just chiming in to say go with ManiacalAce -- he got the correct answer.\n\nCorrect second derivative is as follows:\n\n(-9/4)x^(-3/2) - (3/4)x^(-1/2)\n\n~ ♥ ~
793357
Another way to look at is follows.\n\nThink of d/dx () to mean "differentiate whatever is in the parens with respect to x.  So, for example d/dx (x)=dx/dx=1.   If y=y(x). y'(x)=d/dx (y(x))=dy/dx where the argument of y is omitted for convenience.\n\nSo, d/dx (y')=d/dx (dy/ dx)=(d d y)/(dx dx)=d^2y/dx^2. \n\nThe notation is ambiguous.  One would usually think that the x is being square in the denominator, but "dx" is a single symbol, so in fact, in this context, dx^2=(dx)^2.\n\nI hope this helps!\n\nGood luck. :)
392390
it means you have to derive it twice ,which means you have to derive its second order derivative ,for example derivig the distance formula gives you the speed,then deriving the speed which is the 1st derivative of distance gives you the acceleration.
99360
In kinematics, the 3rd time derivative of position x(t) (x'''(t)) gives the rate of change of acceleration with respect to time (da(t)/dt). It is also known as jerk. This quantity is relevant in the context of problems involving rockets that have a mass that changes with time.
45131
Calculus was confusing enough in English - I can't imagine taking it in Japanese.  You might want to try to get hold of a textbook or something in English.\n\nIt's an integral.  More specifically an indefinite integral.  It's kind of like the inverse of the derivative function except for that there is no way to get the constant back (hence the "C").  Basically the indefinite integral is the set of all functions whose derivative is equal to the thing between the integral sign and the dx (or dy, dq, whatever the variable is... I'm blanking on general name of the thing).  The integral symbol tells you it's an integral (obviously), and the "x" in dx tells you what variable you're integrating with respect to (that will make more sense when you learn to integrate more complicated expressions).  So the indefinite integral of x^2 is 1/3x^3 + C with respect to x where C is some constant because the derivative of 1/3x^3 + C is x^2 (remember that the derivative of a constant is zero).\n\n***A Note on Definite and Indefinite Integrals***\nA couple of the answers below appear to be describing the definite integral.  The definite integral can be described as the sum of the areas of rectangles, and can be used to find the area under a curve (among other things).  The indefinite integral (or antiderivative), however, which is the one refered to in the question, describes a function.  Although the function described by the indefinite integral is extremely useful in calculating the area under a curve, it does not directly describe this area.  In addition to being used to caculate the definite integral, the indifinite integral can also be used, for example, to find an equation for the position of an object given an equation for its velocity.\n\nAnother afterthough: I may have gotten some of the terminology wrong as my high school math department had a strange affinity for inventing their own terms.
916561
This is a fraction, yes?\n\nThe derivative of a fraction is\n\nThe derivative of the numerator times the denominator\nminus\nThe derivative of the denominator times the numerator\nAll divided by the square of the denominator\n\nLet's do it:\n\n[-sin(x) * (1 - cos(x) - sin(x) * (1 + cos(x)] / [1 - cos(x)]^2\n[-2sin(x)] / [1 - cos(x)]^2\n\nNotice that your function is discontinuous at x = 0 and x = pi, etc., and so is the derivative.\n\nI LUV screwed up again. The derivative of cos(x) is -sin(x), not sin(x) as she has it.\n\nSteve's solution agrees with mine.
898061
No.\n\nIt means that the slope of the curve y = x^2 at a given point is equal to twice the value of x at that point.\n\nThe change in y for an infinitesimally small change in the value of x is equal to twice the value of x at that point.\n\nYour interpretation, "for every change in x there is a change of 2 [in y]", would correspond to a derivative (or slope) of 2, not of 2*x.  The function, y = 2*x + constant would have a derivative dy/dx = 2.
278502
Derivative of a function is the study of microscopic and complex rate of change or infinitesimal rate of change of a curve, a process, a phenomena etc.Several changes have been taking place in this world,a macroscopic changes and uniform changes are easily countable, but complex changes are not easily accountable.To understand such changes we require the wonderful mathematical tool  called drivative, diffrentiation, diffrential etc.For example think of pouring  water into a conical vessel in drops.Here if the drops that are dripping in are uniform, it is easy to know the number of drops per minute, per hour etc.But it very difficult to find the rate of change of height of water  as the water drips in to the container (conical) and the corresponding rate of change of surface area of the water in the container.To understand such changes we move towards zero or towards a number which we call as the limit.Also a prosses may be a chemical reaction, in physics it may be a free fall of an object or an acclerated body.Moreover, consider y=x^2, the derivative of this is dy/dx = 2x, here change is not uniform as common difference in an arithmetic progression.\nWhen x=4 & 5 we have x^2= 16 & 25 ; then the difference is 9.Compare this 9 with the derivative 2x = 2*4 & 2*5 = 8 & 10.These numbers 1 more or less to 9.Of course the difference 1 is considerable when x is a small number. When x is sufficiently large this 1 becomes negligible.ie. limit as x tends to infinity the derivative of x^2 is 2x.  \nAnti-derivative is opposite to this concept. This is usually called as integration, summation, premitive etc. Here it is the study of complex accumulation.\nFor example when a point is accumulated in a single direction without break we call it a line, ie. when a point dx is integrated we get a line `x'.When line is intigrated we get a surface, that is x^2 /2. When the same is revesed we get a point . former is called integration later is called diffrentiation.
945648
Take the derivative of f :\nf'(x) = 5x^4 (x+7)^4 + 4 (x+7)^3 x^5\nf'(x)= x^4 (x+7)^3 [ 5 (x+7) + 4x]\nf'(x)= x^4 (x+7)^3 [ 9x + 35]\n\nSo the derivative is equal to 0 for the values :\nx= -7\nx= - 35/ 9\nx = 0\n\nthe function is increasing in [-11;-7],\nthe function is increasing in [-35/9;+12],
858871
let's say its da/dx means that you are taking derivates of a respect to x.
965952
Implicit differentiation is really jsut an exercise in using the chain rule. For instance, if you had e^(x^2), its derivative would be e^(x^2) * 2x, since d(x^2)/dx = 2x. It's the same thing with implicit, except that you don't actually know what dy/dx is yet, so you just write dy/dx. So for my example, d(e^y)/dx = e^y dy/dx. Now, onto your problem:\n\nd(x/(x+y))/dx = ((x+y) - (1+dy/dx)x)/(x+y)², or being simplified, (y-x dy/dx)/(x+y)²\n\nOf course, this assumes you are differentiating with respect to x. If you are differentiating w.r.t. a third variable (e.g. t), then you would have to use the chain rule on x as well. So your derivative would be:\n\nd(x/(x+y))/dt = (dx/dt (x+y) - (dx/dt + dy/dt)x)/(x+y)²
139329
calculus, fun stuff. To get the derivative, you need to drop the exponent then subtract one from the explonent. SO the X^2 would be 2x and 2x would just be 2 .    Final answer is 2x + 2.   Hm.. explained that poorly. In general, to find the derivative, Take the exponent on the X, drop it in front so its multipluing, than subract one from the exponent and have that number in the old exponents place (3x^3 would turn into 9x^2, 4x^2 would be just 8x) any number with X^1 is just the number, and just a number with no X is 0.
498558
Derivatives... (in a nutshell)\n\nAh! This gets a little difficult, because the term derivatives is not really well-defined. It has become a catch-all generic term that has been used to include all types of new (and some old) financial instruments.\n\nHence, the first lesson we must learn from this is not to generalise about derivatives (i.e. not to say, derivatives are: good/bad, risky, volatile etc.). \n\nThe most common types of derivatives that ordinary investors are likely to come across are futures, options, warrants and convertible bonds. Beyond this, the derivatives range is only limited by the imagination of investment banks. It is likely that any person who has funds invested, an insurance policy or a pension fund, that they are investing in, and exposed to, derivatives - wittingly or unwittingly. \n\nA futures contract is an agreement to buy (or sell) some commodity at a fixed price on a fixed date. In other words, it is a contract between two parties; the holder of the future has not only the right but also the obligation to buy (or sell) the specified commodity. This differs sharply from stock options, which carry the right but not the obligation to buy or sell a stock. \n\nThese days, all details of a futures contract are standardized, except for the price of course. These details are the commodity, the quantity, the quality, the delivery date, and whether the contract can be settled in goods or in cash. Futures contracts are traded on futures exchanges, of which the U.S. has eight. \n\n\nFutures...\n\nFutures are commonly available in the following flavors (defined by the underlying "cash" product): \n\nA commodity future, for example an orange-juice future contract, gives you the right to take delivery of some huge amount of orange juice at a fixed price on some date. Alternately, if you wrote (i.e., sold) the contract, you have the obligation to deliver that OJ to someone. \n\nSince you can't really buy an index, these are settled in cash. \n\nInterest rate futures (including deposit futures, bill futures and government bond futures)\nAgain, since you cannot easily buy an interest rate, these are usually settled in cash as well. \nFutures are explicitly designed to allow the transfer of risk from those who want less risk to those who are willing to take on some risk in exchange for compensation. A futures instrument accomplishes the transfer of risk by offering several features: \n\nLeverage (a small amount of money controls a much larger amount)\n \nA high degree of correlation between changes in the futures price and changes in price of the underlying commodity. \nIn the case of the commodity future, if I sell you a commodity future then I am promising to deliver a fixed amount of the commodity to you at a given price (fixed now) at a given date in the future. \nHere are a couple of contrived examples to illustrate how futures might be traded. If the price of the future becomes very high relative to the price of the commodity today, I can borrow money to buy the commodity now and sell a futures contract (on margin). If the difference in price between the two is great enough then I will be able to repay the interest and principal on the loan and still have some riskless profit; i.e., a pure arbitrage (although I might have to pay some storage fees on the commodity). \n\nConversely, if the price of the future falls too far below that of the commodity, then I can short-sell the commodity and purchase the future. I can (presumably) borrow the commodity until the futures delivery date and then cover my short when I take delivery of some of the commodity at the futures delivery date. I say presumably borrow the commodity since this is the way bond futures are designed to work; I am not certain that comodities can be borrowed. \n\nHere are a few resources on futures:\n\nThe Futures FAQ has quite a bit of information:\nhttp://www.ilhawaii.net/~heinsite/FAQs/futuresfaq.html \nThe Futures Industry Associ