Differentiate Variable

On the definition of the derivative is a dynamical diagram displaying the derivative as the slope of the tangent to a graph. The necessary preliminaries one should be familiar with are the slope of a straight line and the graph of a function.

When we differentiate any equation we denote them using different variables .these variables can be one or many depending on the equation. These variables are alphabets like x, y, t, u, v,  etc.


Examples of differentiate variables

There are many examples of differentiation. Such as:

Differentiate xx  + (cos x)1/2 with respect to x

Let y =  xx  + (cos x)1/2

We have + or – sign in between the terms, then we should not tale log.

We should write y = u + v

dy/dx = du/dx + dy/dx                                            …..(i)

consider  u = xx

taking log on both sides, we get

log u = x. log x

differentiating with respect to x, we get

1/u . du/dx = x.d/dx (log x) + log x.dx/dx

1/u. du/dx = x.1/x + log x.1

du/dx = u(1 + log x) = x2( 1 + log x)                 …..(ii)

again consider  v = (cos x)1/x

taking log on both sides, we get

log v = 1/x.log(cos x)

differentiating with respect x, we get

1/v.dv/dx = 1/x.d/dx log (cos x) + log(cos x).d/dx(1/x)

1/v.dv/dx = 1/x. 1/ cosx. (-sin x) + log(cos x). (-1/x2)

dv/dx = v[- tanx /x – 1/x2. Log (cos x)]

= (cos x)1/x[- tan x/x – 1/x2.log(cos x)]                    ……(iii)

Substituting from (ii) and (iii) in (i), we get

dy/dx = x2(1+ log x) + (cos x)1/x [-tanx/x – 1/x2. Log(cos x)]

here the variable used are x, y , u and v, and these are called the differentiate variables.


Problems on differentiate variables

Differentiate , y = (x)cos x + (sin x)tan x
y= x.sin x.log x.