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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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