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Example:Logarithmic Differentiation

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In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity (conversely, integrating a car's velocity over time yields the distance traveled).

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function.

The process of finding a derivative is called differentiation.

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 respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + b, for real numbers m and b, and the slope m is given by

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in."




Differentiation of trigonometric functions


The differentiation of trigonometric functions is the mathematical process of finding the rate at which a trigonometric function changes with respect to a variable--the derivative of the trigonometric function. Commonplace trigonometric functions include sin(x), cos(x) and tan(x). For example, in differentiating f(x) = sin(x), one is calculating a function f ′(x) which computes the rate of change of sin(x) at a particular point a. The value of the rate of change at a is thus given by f ′(a). Knowledge of differentiation from first principles is required, along with competence in the use of trigonometric identities and limits. All functions involve the arbitrary variable x, with all differentiation performed with respect to x.






− sin(x)




− csc2(x)




− csc(x)cot(x)





(Our solved example in mathguru.com uses this concept)



















The logarithm of a number y with respect to base b is the exponent to which b has to be raised to yield y. In other words, the logarithm of y to base b is the solution x of the equation

The logarithm is denoted "logb(y)" (pronounced as "the logarithm of y to base b" or "the "base-b logarithm of y"). In the equation logb(y) = x, the value x, is the answer to the question "To what power must b be raised, in order to yield y?"For the logarithm to be defined, the base b must be a positive real number not equal to 1 and y must be a positive number



Analytic properties of functions pass to their inverses. Thus, as f(x) = bx is a continuous and differentiable function, so is logb(y). Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f(x) evaluates to ln (b) bx by the properties of the exponential function, the chain rule implies that the derivative of logb(x) is given by

That is, the slope of the tangent touching the graph of the base-b logarithm at the point (x, logb(x)) equals 1/(xln(b)). In particular, the derivative of ln(x) is 1/x, which implies that the antiderivative of 1/x is ln(x) + C. The derivative with a generalised functional argument f(x) is

(Our solved example in mathguru.com uses this concept)






The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.