Differentiation

Derivative of function y =f(x) at point x is,

\[\displaystyle {{f}^{'}}(x)=h\overset{{\lim }}{\longrightarrow}0\frac{{f(x+h)-f(x)}}{h}\]

\[\displaystyle =\Delta x\overset{{\lim }}{\longrightarrow}0\frac{{\Delta y}}{{\Delta x}}=\frac{{dy}}{{dx}}----(1)\]

Here, ∆x = h and ∆y =f(x+h) – f(x) are the increments in the variables x and y.

Derivative of f(x) is denoted by dy/dx is the limit is,

\[\displaystyle \frac{{dy}}{{dx}}=h\overset{{\lim }}{\longrightarrow}0\frac{{f(x+h)-f(x)}}{h}----(2)\]

It is interpreted as the rate of change of y with respect to x. The process of finding a derivative is called differentiation.

Derivative of the function f(x) at a given point represents slop of the tangent draw the curve y = f(x) at the point where the function is defined.

DIFFERENTIATION FORMULA

A few differentiation formula is given below; these formula assume that u and v are a differentiable function of x and c, and n are arbitrary constant.

\[\displaystyle \frac{d}{{dx}}(c)=0\]

\[\displaystyle \frac{d}{{dx}}(u+v)=\frac{{du}}{{dx}}+\frac{{dv}}{{dx}}\]

\[\displaystyle \frac{d}{{dx}}c(x)=c\frac{{du}}{{dx}}\text{ (constant multiple rule)}\]

\[\displaystyle \frac{{d(uv)}}{{dx}}=u\frac{{dv}}{{dx}}+v\frac{{du}}{{dx}}\text{ (product rule)}\]

\[\displaystyle \frac{d}{{dx}}\left( {\frac{u}{v}} \right)=\frac{{v\frac{{du}}{{dx}}-u\frac{{dv}}{{dx}}}}{{{{v}^{2}}}},v\ne 0\text{ (quotient rule)}\]

\[\displaystyle \frac{{d{{x}^{n}}}}{{dx}}=n{{x}^{{n-1}}}\text{ (power rule for positive and negative integers)}\]

\[\displaystyle \frac{{d{{e}^{x}}}}{{dx}}={{e}^{x}},\frac{{d{{a}^{x}}}}{{dx}}={{a}^{x}}\log a\]

\[\displaystyle \frac{d}{{dx}}\log (x)=\frac{1}{x},\frac{d}{{dx}}\sin (x)=\cos x\]

MAXIMA AND MINIMA

Maxima and minima is a graph of functions described in some cases; the function is increasing or decreasing.

The first derivative, f’(x) and the second derivative, f’’(x), is a function f(x) that plays a main role here. Function f(x) is said to have maxima for,

\[\displaystyle x=a\text{ if }f(x)-f(a)=-ve\text{ for all }x,\text{when}\left| {x-a} \right|\text{ }\langle \text{ }\varepsilon \]

Here, ε = small positive number

A function f(x) has a minimum for x= a if,

\[\displaystyle x=a\text{ if }f(x)-f(a)=-ve\text{ for all }x,\text{when}\left| {x-a} \right|\text{ }\langle \text{ }\varepsilon \]

Here, ε = small positive number

The figure shows that the f(x) →x graph shows several maxima and minima, from the figure that a, c and e are the maxima and b and d are the minima.

It pointed out that the maximum value of f(x) at a point is not necessarily its greatest value; the same, the minimum value of the function at a point is not necessarily its smallest value. The following points should be noted;

One minimum or maximum lies between two equal values of a function.
Maxima and minima must occur alternatively, cone after the other.
A function f(x) may have various maxima and minima.
Function y =f(x) is maximum at x = a if dy/dx changes its sign from +ve to -ve as x passes through a.
Function y = f(x) is minimum x = a if dy/dx changes sign from -ve to +ve as x passes through a.
If the sign of dy/dx does not change while x passes through a, then function y = f(x) is neither maximum nor minimum at x = a.
If f’(x) > 0, a function f(x) is monotonically increasing.
If f’ (x) < 0, a function f(x) is monotonically decreasing.

Maxima and minima f(x)

RULES FOR FINDING MAXIMA- MINIMA

Find f’(x)
Solve f’(x) = 0. Each value of x obtained is a candidate for maximum or minimum.

Let x= c become its points.

Find f’ (x)
x = c is point minima, where f’’ (c) >0.
x = c is a point maxima, if f’’ (c) < 0.

Maxima and minima in an internal, f(x) be a function defined in an interval maximum. Then, maximum { f(a), f(b), f(c)} is the maximum value of the function.

Same, let x = d be a point of a minimum, then the minimum

{ f(a), f(b), f(d)} is the minimum value of the function.

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