Presentation Cvt Sameer

8/8/2019 Presentation Cvt Sameer

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In calculus, a branch ofmathematics, the

derivative is a measure of how a function

changes as its input changes.


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` Differentiation is a method to compute the rate at

which a dependent output y changes with respect

to the change in the independent input x. This rate

of change is called the derivative of y with respectto x.


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` This function does not have a derivative at the

marked point, as the function is not continuous

there.


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` The absolute value function is continuous, but fails

to be differentiable at x = 0 since the tangent

slopes do not approach the same value from theleft as they do from the right.


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` Leibniz's notation

` Lagrange's notation

` Newton's notation

` Euler's notation


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` Derivatives of powers: if

` where r is any real number, then


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` Sum rule:

` Product rule:

` Quotient rule:

` Chain rule:


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I

In general, the partial derivative of a function (x1, , xn) in the direction xi at the point (a1 , an) is defined to be:


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` In mathematics, an analytic function is a function

that is locally given by a convergent power series.

` Analytic functions can be thought of as a bridge

between polynomials and general functions.` There exist both real analytic functions and

complex analytic functions.


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` Formally, a function is real analytic on an open

set D in the real line if for any x0 in D one can write

in which the coefficients a0, a1, ... are real

numbers and the series is convergent to (x) for x

in a neighborhood of x0.


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` Any polynomial (real or complex) is an analytic

function

` The exponential function is analytic` The trigonometric functions, logarithm, and the

power functions are analytic


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` The absolute value function when defined on the

set of real numbers or complex numbers is not

everywhere analytic because it is not differentiable

at 0` The complex conjugate function is not complex

analytic, although its restriction to the real line is

the identity function and therefore real analytic.


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` The sums, products, and compositions of analytic

functions are analytic.

` The reciprocal of an analytic function that is

nowhere zero is analytic, as is the inverse of aninvertible analytic function whose derivative is

nowhere zero

` Any analytic function is smooth, that is, infinitely

differentiable.


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` As noted above, any analytic function (real or

complex) is infinitely differentiable (also known as

smooth, or C). (Note that this differentiability is in

the sense of real variables; compare complexderivatives below.) There exist smooth real

functions which are not analytic: see non-analytic

smooth function.


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Presentation Cvt Sameer