chứ đạo hàm rồi mới nguyên hàm thì không ổn
$f(x)\not\equiv \int f\;'(x)\;\mathrm dx \not\equiv \int \mathrm d(f(x))$
xin sửa lại 1 chút f(x) + C = $\int f\;'(x)\;\mathrm dx =\int \mathrm d(f(x))$
thì vẫn ổn
06-07-2012 - 12:37
chứ đạo hàm rồi mới nguyên hàm thì không ổn
$f(x)\not\equiv \int f\;'(x)\;\mathrm dx \not\equiv \int \mathrm d(f(x))$
06-07-2012 - 07:32
06-07-2012 - 06:57
06-07-2012 - 06:36
The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs
The integral sign ∫ represents integration. The dx indicates that we are integrating over x; dx is called the variable of integration. In correct mathematical typography, the dx is separated from the integrand by a space (as shown). Some authors use an upright d (that is, dx instead of dx). Inside the ∫...dx is the expression to be integrated, called the integrand. In this case the integrand is the function f(x). Because there is no domain specified, the integral is called an indefinite integral.
06-07-2012 - 00:30
compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.
In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written
dy / dx
suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of y with respect to x", "d y by d x", or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)
This expression is Newton's difference quotient. The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function f at a is the limit
of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then f is differentiable at a. Here f′ (a) is one of several common notations for the derivative (see below).
Equivalently, the derivative satisfies the property that
which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation
to f near a (i.e., for small h). This interpretation is the easiest to generalize to other settings (see below).
Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly using this method. Instead, define Q(h) to be the difference quotient as a function of h:
Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from h = 0. If the limit exists, meaning that there is a way of choosing a value for Q(0) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q(0).
In practice, the existence of a continuous extension of the difference quotient Q(h) to h = 0 is shown by modifying the numerator to cancel h in the denominator. This process can be long and tedious for complicated functions, and many shortcuts are commonly used to simplify the process.
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