DERIVATIVES
Department of Mathematics
Uniglobe higher secondary school
Kamaladi Ganeshthan, Kathmandu, Nepal
Submitted by:
Aakriti Shrestha
Under supervision of: Mr. Agni Datta Joshi
Abstract
Let y=f(x) be a function defined as the interval of (a, b). Them
the function is said to be differentiable if the limit
h lim → 0 [f(x + h) –f(x)] / h exists
If this limit exists, then it is called the derivative of f(x). It is
denoted by fᶦ (x). It is denoted by fᶦ (x) or yᶦ = dy / dx.
Derivatives are a fundamental tool of calculus. For example,
the derivative of the position of a moving object with respect
to time is the object's velocity. The derivative of a function of
a single variable at a chosen input value, when it exists, is the
slope of the tangent line to the graph of the function at that
point. The process of finding a derivative is
called differentiation. The reverse process is
called antidifferentiation. The fundamental theorem of
calculus relates antidifferentiation with integration.
Table of content
Introduction to derivative
Some important formulas
Hyperbolic function formulas
Conclusion
References
Introduction
HISTORY:
The problem of finding the tangent to a curve has been studied
by many mathematicians since Archimedes explored the
question in Antiquity. The first attempt at determining the
tangent to a curve that resembled the modern method of the
Calculus came from Gilles Persone de Roberval during the
1630's and 1640's. At nearly the same time as Roberval was
devising his method, Pierre de Fermat used the notion of
maxima and the infinitesimal to find the tangent to a curve.
Some credit Fermat with discovering the differential, but it was
not until Leibniz and Newton rigorously defined their method
of tangents that a generalized technique became accepted.
Introduction:
In mathematics, the derivative is the exact rate at which one
quantity changes with respect to another. Geometrically, the
derivative is the slope of a curve at a point on the curve,
defined as the slope of the tangent to the curve at the same
point. The process of finding the derivative is called
differentiation. This process is central to the branch of
mathematics called differential calculus.
In mathematical terms: h lim → 0 [f(a+b) –f(a)] / h
Limit:
A function f (x) is said to have limit A at point x=α, if given a
small positive number ε, there exists another positive number
§ such that whenever 0 < | x –α |< §, we have | f(x) –A | < ε.
When x approaches α within a distance of §, f (x) approaches
A within a distance of ε from above (right side of α) we write
x lim→α+ f(X) = A or h lim→0 f(a + h) = A
Similarly if we replace 0<|x-α|<§, we say that lim f(x) =A as x
approaches α from below (left side of α) we write
x lim→α- f(x) =A or h lim→0 f(a – h) =A
The limits x lim→α+ f(x) and x lim→α- f(x) are called right
hand limit and left hand limit respectively of f(x) at x = α.
Thus, we can say a function has limit only if left hand limit is
equal to the right hand limit.
Continuity:
A function is f(x) is said to be continuous at x = a, if x lim→α
f(x) = f(α).
This definition shows that the limit x lim→α f(x) must exist if
for f(x) to be continuous. But it may happen that though x
lim→α f(x) exits, the value of x lim→α f(x) may not be equal
to f(α). In that case, f(x) is not continuous at f(α). So we can
say that the existence of limit f(x) at x=α is necessary but is
not the sufficient condition for the continuity of function f(x)
at x=α. Continuity of functions depend on how we define the
functions at x = 0.
Some important formulas
Basic formula:
1. x lim→ 0 eͯ -1/x = 1
2. x lim → 0 aͯ -1/x = logx
3. x lim →0 log(1+x)/x = 1
Derivatives of trigonometric function
1. d/dx ( sin x ) = cos x
2. d/dx ( cos x) = - sin x
3. d/dx ( tan x ) = sec² x
4. d/dx (cosec x) = - cosec x * cot x
5. d/dx (sec x ) = sec x * tan x
6. d/dx (cot x ) = - cosec²x
Hyperbolic function formulas
Sinhx, coshx , tanhx , coseechx , sechx , cothx are hyperbolic
functions.The hyperbolic functions for real number are
defined as:
1. Sinhx =
2. Coshx =
3. Tanhx =
4. Cosechx =
5. Sechx =
6. Cothx =
Derivatives of hyperbolic function:
1. d/dx ( sinhx ) = coshx
2. d/dx ( coshx ) = sinhx
3. d/dx ( tanhx ) = sec²hx
4. d/dx (cosechx ) = -cosechx * cothx
5. d/dx (sechx ) = -sechx * tanhx
6. d/dx ( cothx) = - cosec²hx
Derivatives of inverse hyperbolic function:
1. d/dx sin̄¹ =
2. d/dx cos̄¹ =
3. d/dx tan̄¹ =
4. d/dx cosec̄¹ =
5. d/dx sec̄¹ =
6. d/dx cot̄¹ =
Conclusion
Differentiation is the action of computing a derivative. The
derivative of a function y = f(x) of a variable x is a measure of
the rate at which the value y of the function changes with
respect to the change of the variable x. It is called the
derivative of f with respect to x. the derivative is often
described as the "instantaneous rate of change", the ratio of
the instantaneous change in the dependent variable to that
of the independent variable.
Derivatives can be generalized to functions of several real
variables. In this generalization, the derivative is
reinterpreted as a linear transformation whose graph is (after
an appropriate translation) the best linear approximation to
the graph of the original function.
Reference
Britannica.com visited on November,3
Wikepedia.org visited on November,3
Basic mathematic Grade XII
