1. Miami Dade College -- Hialeah Campus. Calculus I Formulas. MAC 1. Limits and Derivatives. 2. Differentiation rules. 3. Applications of Differentiation. 4. Elementary Differential and Integral Calculus. FORMULA SHEET. Exponents xa · xb = xa+b, ax · bx = (ab)x, (xa)b = xab, x0 = 1. Logarithms lnxy = lnx + lny, lnxa. Differentiation Formulas d dx k = 0. (1) d dx. [f(x) ± g(x)] = f (x) ± g (x). (2) d dx. [k · f (x)] = k · f (x). (3) d dx. [f(x)g(x)] = f(x)g (x) + g(x)f (x) (4) d dx. (f(x) g(x).) = g(x)f (x).
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in Chapter X. of the Differential Calculus, on Maxima and Minima, and in Chapter .. whose Equation is given between Three Rectangular Co- ordinates, jc, y, z. Use differentiation and integration tables to supplement differentiation and integration techniques. Differentiation Formulas. 1. 2. 3. 4. 5. 6. 7. 8. 9. Vector Calculus. . Finding the zeros of equations; Numerical integration of differential Speigel, M.R., Mathematical Handbook of Formulas and Tables.
This yields the general form of the Leibniz integral rule,. Now, set. Then, by properties of Definite Integrals , we can write. There is a technical point in the proof above which is worth noting: For a rigidly translating surface, the limits of integration are then independent of time, so:.
This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface.
Having found the derivative, variables can be switched back to the original frame of reference.
Differential calculus basic formulas pdf
We notice that see article on curl. The sign of the line integral is based on the right-hand rule for the choice of direction of line element d s.
Consequently, the sign of the line integral is taken as negative. This proof does not consider the possibility of the surface deforming as it moves. From proof of the fundamental theorem of calculus ,.
If one defines:. By the Heine—Cantor theorem it is uniformly continuous in that set.
The principle of differentiating under the integral sign may sometimes be used to evaluate a definite integral. This is somewhat inconvenient. There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign.
The measure-theoretic version of differentiation under the integral sign also applies to summation finite or infinite by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence. Differentiation under the integral sign is mentioned in the late physicist Richard Feynman 's best-selling memoir Surely You're Joking, Mr.
He describes learning it, while in high school , from an old text, Advanced Calculus , by Frederick S. Woods who was a professor of mathematics in the Massachusetts Institute of Technology. The technique was not often taught when Feynman later received his formal education in calculus , but using this technique, Feynman was able to solve otherwise difficult integration problems upon his arrival at graduate school at Princeton University:. One thing I never did learn was contour integration.
I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again. I was up in the back with this book: Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college.
It had Fourier series , Bessel functions , determinants , elliptic functions —all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it.
But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school.
If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked.
So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me. From Wikipedia, the free encyclopedia. This article is about the integral rule. For the convergence test of alternating series, see Alternating series test.
This article needs additional citations for verification. A general function is not a line, so it does not have a slope.
Since the derivative is the slope of the linear approximation to f at the point a, the derivative together with the value of f at a determines the best linear approximation, or linearization , of f near the point a. If every point a in the domain of f has a derivative, there is a function that sends every point a to the derivative of f at a.
A closely related notion is the differential of a function. When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number.
MATH6103 & MATH6500
If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. The linearization of f in all directions at once is called the total derivative. History of differentiation[ edit ] Main article: History of calculus The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid c.
The use of infinitesimals to study rates of change can be found in Indian mathematics , perhaps as early as AD, when the astronomer and mathematician Aryabhata — used infinitesimals to study the orbit of the Moon.
Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes,  which had not been significantly extended since the time of Ibn al-Haytham Alhazen.
Regarding Fermat's influence, Newton once wrote in a letter that "I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general. Since the 17th century many mathematicians have contributed to the theory of differentiation.
In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy — , Bernhard Riemann — , and Karl Weierstrass — It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.
If f is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points. This is called the second derivative test.He describes learning it, while in high school , from an old text, Advanced Calculus , by Frederick S. If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:.
In particular, the time derivatives of an object's position are significant in Newtonian physics : velocity is the derivative with respect to time of an object's displacement distance from the original position acceleration is the derivative with respect to time of an object's velocity, that is, the second derivative with respect to time of an object's position. The clustrmap is periodically and automatically archived and its counters reset, so the total is smaller. Due to the nature of the mathematics on this site it is best views in landscape mode.
A general function is not a line, so it does not have a slope. In each of these regions we know that the derivative will be the same sign. Differentiation under the integral sign is mentioned in the late physicist Richard Feynman 's best-selling memoir Surely You're Joking, Mr. See also: