what are derivatives?
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A derivative is a function measuring the rate of change of another function. A common example is velocity. Velocity is the derivative of position, because velocity is measuring the rate of change of position. And acceleration is the derivative of velocity, because the rate of change of velocity is acceleration.\n\nIt's the basic cornerstone of calculus.
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well if its dealing with derivitives\n\nf(x) = x^2\nf'(x) = 2x^1\n\nor the general rule\nf(x) = x^n\nf'(x) = n(x^(n-1))
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If one has a function f as a map from the real line into itself, one also has the graph of the function as the set of all points (x,f(x)) as a curve in the Cartesian plane. If one wanted to know the area between the x-axis and this curve over some interval on the x-axis, one would use a definite integral. If one wanted to know the slope of a line tangent to the curve at some point (x,f(x)) on the curve, one would use the derivative of the function f at the point x. Both of these operators are very useful in the sciences.\n \nFor differentiation in one dimension, you can use the definition: the derivative of a function f at the (domain) point x is the limit as h approaches 0 of the slope of the secant line [f(x+h)-f(x)]/h. Visually, we take a secant line to the curve that passes through the points (x, f(x)) and (x+h, f(x+h)) and rotate it (see what happens as h gets smaller) until it only touches the curve once (it is now tangent to the curve). This line is unique at each point.\nIn more than one dimension, using the definition is cumbersome for anything other than proving when a derivative does or does not exist, so we use a useful theorem that says the derivative of a function f from R (the real line) into R^n (n-dimensional Euclidean space) is just the vector of partial derivatives of f. Ie., if f(x,y) = x^2 + y^2 (the graph is a circular paraboloid), then Df(x,y) = (df/dx, df/dy) = (2x, 2y). Partial derivatives are taken relative to one variable only; they end up treating the other variables as constants. The technical definition of the partial of f with respect to x at the point x0 is what you suspect it would be: the limit as h approaches zero of the slope of the secant line [f(x0+h, y) - f(x0,y)]/h.\nOnce one gets to these derivatives of multivariable functions though, one must be very careful, as the existence of all partial derivatives at a point does not mean the derivative exists at that point.\nOnce one gets a bit of experience taking derivatives using the definition, one can derive some simple derivative rules which allow you to take derivatives of elementary functions without explicitly doing all that limit mumbo-jumbo. A list of common differentiation rules can be found here: http://mathworld.wolfram.com/Derivative.html . \n\nIntegration is a more delicate matter. If you're doing a definite Riemann integral (finding the area under a curve), it is the limit of a Riemann sum as you increase the number of rectangles under the curve ( See http://mathworld.wolfram.com/RiemannSum.html ). The indefinite integral is just the function that will return the definite integral when you take the difference between two points. \nSometimes the rectangles will not converge to any finite value or it does not have a unique convergence and we say the function is not integrable on that interval (Take f(x) = 1/x on the interval [0,1] for example). A more technical definition of integration will help integrate more pathological functions, but you can look up this definition yourself in a real analysis text; it's not that different from summing rectangles.\nAgain, with integrals we have a very nice theorem that tells us that the indefinite integral of the derivative of a function f is f itself. This is the fundamental theorem of calculus, and it takes a lot of work out of finding an integral. For example, if f(x) = 2x, we know from our experience with taking derivatives that the indefinite integral of f(x) is just F(x) =  x^2 + C where C is an unknown constant since the derivative of any such F is 2x. The indefinite integral is thus sometimes called the antiderivative in older texts. If we want to know the area under the curve f(x)=2x above the interval [0, 5], we need only evaluate  F(5) - F(0) = 25. Of course, we will want to find the area under more advanced curves, but this gave a simple example.\nAs there are derived rules for derivation, there are also rules for integration, although most are just rearranged rules of diffe
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You use derivatives to compute how values are changing. A good example is to compute velocities.\n\nIntegration is used to compute areas and volumes.
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The first derivative is the slope of the line that is tangent to the graph.  If there is a maximum or minimum, the tangent line at that point will be horizontal, so its slope is 0.  \n\nThe second derivative is 0 when there is an inflection point. An inflection point is a place on the graph where it changes the direction of the concavity. \n\nIn the two cases above, the converse is not necessarily true.  That is, just because the first derivative is 0 doesn't mean there has to be a max/min there.\n\n If the second derivative is positive, the graph is concave upward which means like a regular "U".  If it's negative, it's concave downward like an upside-down "U".
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This is a Frequently Asked Question. Following is a modified previous answer:\n\n\nIf one has a function f as a map from the real line into itself, one also has the graph of the function as the set of all points (x,f(x)) as a curve in the Cartesian plane. If one wanted to know the area between the x-axis and this curve over some interval on the x-axis, one would use a definite integral. If one wanted to know the slope of a line tangent to the curve at some point (x,f(x)) on the curve, one would use the derivative of the function f at the point x. Both of these operators are very useful in physics, as most physical properties are defined as changes in some property. If p(x) is a graph of some physical property over the domain it is a function of, then the change in p with respect to x approaches the slope of the tangent line to p(x) as one considers smaller and smaller changes in x. Thus, the derivative is the ideal way of talking about continuous changes in physical properties.\n\nFor differentiation in one dimension, you can use the definition: the derivative of a function f at the (domain) point x is the limit as h approaches 0 of the slope of the secant line [f(x+h)-f(x)]/h. Visually, we take a secant line to the curve that passes through the points (x, f(x)) and (x+h, f(x+h)) and rotate it (see what happens as h gets smaller) until it only touches the curve once (it is now tangent to the curve). This line is unique at each point.\nIn more than one dimension, using the definition is cumbersome for anything other than proving when a derivative does or does not exist, so we use a useful theorem that says the derivative of a function f from R (the real line) into R^n (n-dimensional Euclidean space) is just the vector of partial derivatives of f. Ie., if f(x,y) = x^2 + y^2 (the graph is a circular paraboloid), then Df(x,y) = (df/dx, df/dy) = (2x, 2y). Partial derivatives are taken relative to one variable only; they end up treating the other variables as constants. The technical definition of the partial of f with respect to x at the point x0 is what you suspect it would be: the limit as h approaches zero of the slope of the secant line [f(x0+h, y) - f(x0,y)]/h.\nOnce one gets to these derivatives of multivariable functions though, one must be very careful, as the existence of all partial derivatives at a point does not mean the derivative exists at that point.\nOnce one gets a bit of experience taking derivatives using the definition, one can derive some simple derivative rules which allow you to take derivatives of elementary functions without explicitly doing all that limit mumbo-jumbo. A list of common differentiation rules can be found here: http://mathworld.wolfram.com/Derivative.html .\n\nIntegration is a more delicate matter. If you're doing a definite Riemann integral (finding the area under a curve), it is the limit of a Riemann sum as you increase the number of rectangles under the curve ( See http://mathworld.wolfram.com/RiemannSum.html ). The indefinite integral is just the function that will return the definite integral when you take the difference between two points.\nSometimes the rectangles will not converge to any finite value or it does not have a unique convergence and we say the function is not integrable on that interval (Take f(x) = 1/x on the interval [0,1] for example). A more technical definition of integration will help integrate more pathological functions, but you can look up this definition yourself in a real analysis text; it's not that different from summing rectangles.\nAgain, with integrals we have a very nice theorem that tells us that the indefinite integral of the derivative of a function f is f itself. This is the fundamental theorem of calculus, and it takes a lot of work out of finding an integral. For example, if f(x) = 2x, we know from our experience with taking derivatives that the indefinite integral of f(x) is just F(x) = x^2 + C where C is an unknown constant since the deriv
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A differential is your "delta-x"... x2-x1.  A derivative is the limit, as the differential in x approaches 0, in the differential in y.   \nlim(delta x->0) of [f(x+delta x) - f(x)]/delta x.\nYou can't write a derivative without the limit, by definition.
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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.
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Instruments, such as options and futures contracts, which derive their value from the value of an underlying security, group of securities or an index.\n\nFinancial instruments, such as futures and options, which derive their value from underlying securities including bonds, bills, currencies, and equities\n\n\nA specialized security or contract that has no intrinsic overall value, but whose value is based on an underlying security or factor as an index. A generic term that, in the energy field, may include options, futures, forwards, etc.
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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
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Partial derivatives are used in any application where the result changes due to more than one variable.\n\nThey are used extensively in fluid mechanics since the velocity, temperature, and density of a fluid usually vary in relation to your position in the flow in all three directions, and with respect to time.\n\nSimilarly, electric and magnetic fields will vary in three dimensions. \n\nThey are also used in theory of elasticity, and strengths of materials. Basically, any application that requires the analysis of a two or three dimensional space will need partial derivation.
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differentiation is the process of finding the rate of change of a gradient.\n( ^= to the power)\n\nsay you are given an equation y=x^n\ndo differentiate, take the power of x, and multiply it by its coefficient. in this case, x has a coefficient of 1. \ny=nx^n\nthe next step it to subtract 1 from the base of x (its power)\ny=nx^n-1\ndifferentation is now complete\n\ne.g.\ny=3x^4\ndy=12x^3\nno matter what the power of x is, just multiply it by the coefficient of x, and subtract 1 from it. (do not subtract then multiply)\n\nIntergration is the opposite, and it too complicated fot this post. talk to you teacher, of go to www.s-cool.co.uk
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phyzikzteacher is right, as well as the next answer after his.  Cusps and corners are notorious points where a function fails to have a SINGLE derivative.  In fact, cusps and corners have an infinite number of tangent values at those points.  The function, however, is differentiable, just that you get a zero in the denominator usually at the x value of the cusp/corner.
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There's a derivatives tutorial on the following page:\n\nhttp://www.karlscalculus.org/calculus.html#s4
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Examples of derivatives include options, futures, and forwards.\n\nDerivatives evolved from a need to negotiate a price for a future delivery date on farmers crops. The earliest commodity producers: corn, wheat, tulips, etc, used forwards to strike deals for their crops before they had been harvested.\n\nToday, there is a wide spectrum of participants in derivatives markets. Speculators, Hedgers, Investors, Producers, Consumers.\n\nUnless you are a corn farmer, or drill oil in your backyard, I would stick with stocks, bonds and mutual funds for at least ten years before even considering derivatives. They carry significant risks.\n\nA person could make many millions of dollars and never trade a derivative.\n\n(Technically, a purchase agreement on a house is a forward contract. This might be the one big derivative trade for most people.)
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Not quite sure what you mean, but I think this is what you're asking for:\n\nif you have y = x^3 + 2x^2 - x + 5\n\nto differentiate that, let's rewrite the equation like this: \n\ny = x^3 + 2x^2 - x^1 + 5x^0\n\nNow, for each item in the equation, multiply that item by its exponent, and subtract 1 from the exponent, giving you:\n\ny' = 3*(x^2) + 2*(2x^1) - 1*(x^0) + 0*(5x^-1)\n\nwhich simplifies to:\n\ny' = 3x^2 + 4x - 1\n\nAnd there you have the derivative of a simple polynomial equation.
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In basic calculus, we use the derivative to find the slope of a curve, and the integral to find the area under a curve.  The question is, why are these operations the opposite of each other?\n\nThe quick answer is:  The derivative is used to calculate the rate of change of a function.  And the rate of change of the area under a curve is actually the original function.\n\n\nLet's say you have a simple curve, y=f(x).\n\nThe derivative of the function is gives you the rate of change of that function.  As x increases by 1, how much does y change?  That's also known as the slope.\n\n[And you probably know that the derivative of the derivative is also useful.  That's the rate of change of the slope.  When the slope stops changing (or its rate of change=0), it's at a maximum, minimum, or point of inflection.]\n\nThe same relationship exists for the area under a curve and the function itself.  You probably learned to think of the integral of f(x) as the sum of the thin vertical strips under the curve.  That's the area under the curve.  As you move to the right, how fast does the area grow?  You're essentially adding these vertical strips to the area.  The length of a vertical strip is f(x), or the original function.  So the original function f(x) represents how fast the area grows for any value x.\n\n\nETA: What do I have to do to get selected as best answer?