What is differentiation and integreation in physics?
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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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Example:\nIf z=f(u,v),\nu=g(x,y) and\nv=h(x,y), then\n(all derivatives are supposed to be partial derivatives)\ndz/dx=dz/du*du/dx+dz/dv*dv/dx\ndz/dy=dz/du*du/dy+dz/dv*dv/dy\n\nAlternatively, you can set of the matrices\n[[dz/du dz/dv]]\nand\n[[ du/dx du/dy]\n[dv/dx dv/dy]]\nand get that\n[[dz/dx dz/dy]]\nis the product of the two matrices.
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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.
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The essence of calculus is the derivative. The derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point. Let's use the view of derivatives as tangents to motivate a geometric definition of the derivative
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A derivative is the primary mathematical tool that is used to calculate and study rates of change.\n\nf'(Xo)=lim f(x+h)-f(x) / f  (/is divide which I am sure u know)\n          \n\nK so thats really confusing-not easy trying to write out the formula on this but hope it helps a little.
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Differentiation and integration are mathematical operations applied to polynomial functions.\n\nThe first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behavior of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance travelled, or volume displaced. These two processes act inversely to each other, as shown by the fundamental theorem of calculus.\n\nThese operations are frequently used solve complex scientific problems, for instance, in the physics area. fundamental advances in calculus during the 17th and 18th century had a deep impact on the ensuing development of physics. Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimal solution to a problem that can be given in mathematical form.
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Let u=ln t, e^u=t, du=dt/t, dt=t du\nThen:\n∫ln² t dt = ∫t u² du = ∫u² e^u du\nIntegrating by parts:\nu² e^u - ∫2u e^u du\nu² e^u - 2u e^u + ∫2e^u du\nu² e^u - 2u e^u + 2e^u + C\nt ln² t - 2t ln t + 2t +C\nAnd we are done.
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well Y' = dy/dx\nso using seperation of veriables we can get\ndy/dx = K - 1/3*y^3\nand dy/(K-1/3*y^3)=dx now integrate both sides. \nDoes that help?
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look up differentiation in your textbook
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i'm not quite sure what you mean, but if you take an ODE, say, for example, the exponential decay formula (dx/dt) + kx = 0, you can write it as dx = -kxdt, and you can solve using separation of variables.  is this what you mean?
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2nd Law of motion:  the net force = the time derivative of the momentum vector.
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Expressions such as yours is called a power tower. They are differentiable by writing them as exponentials but they are not integrable in closed form. You can find the series form at the source link however.
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Depends on what variable you're differentiating with respect to:\ndV/dr : 2/3*pi*r*h\ndV/dh:  1/3*pi*r^2
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Integration is one of the two major operations of calculus (the other being the inverse, differentiation. There are indefinite and definite types of integration. When the integral of a function is expressed symbolically (as another function), this is the indefinite integral (e.g., the ind. int. of y=2x is y=x^2 + a constant). When the integral is numerically evaluated over some range of x, this is the definite integral. Much of time-domain simulation consists of successive evaluation and accumulation of the definite integral of some function, or "time stepping". Typically, each time step starts with evaluation of the 2nd derivative of something, let's say position (whose 2nd deriv. is acceleration), accumulation of the 1st deriv. (velocity) as the previous value + time-step size * 2nd deriv., then accumulation of the 0th deriv. (position) as the previous value + time-step size * 1st deriv. Of course it's a bit more complex than that, and many integration formulas exist to minimize errors.\nAn "integration" function could be something like what I've described for time-stepping, or it could be a much more sophsticated function that accepts a function definition and gives you the indefinite integral in symbolic form. So you need more information about what kind of integration function is desired.
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You would need to know how to integrate (that is, find anti-derivatives). The proceedure forthis type of problem can usually be understood after first semester calculus. The specific integrations may need some second semester calculus.
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let 1+sec x = t\n\ndifferentiate\n\nsec x tan x dx = dt\n\nnow integral = integral of dt/t = ln | t| + c = ln | sec x +1 | + c
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The total differential is [∂F/∂x]dx +[ ∂F/∂y]dy\n\n∂F/∂x = 4x^3*y^-1\n∂F/∂y = x^4*(-y^-2) = -x^4y^-2