The technique of differentiating a function defined parametrically is also applicable in finding the derivative of one function with respect to another function. For example, suppose we wish to find the derivative of f (x) with respect to g (x). Then by letting.

y = f (x), z = g (x)

and treating x as the parameter, we have

dv = dy/dx = f ‘(x) 
dz   dz/dx     g '(x)

Example . Differentiate

(i)    (logx)² with respect to e^x .

(ii)    x² + log x + e^-x with respect to √x.

Solution. (i) Let y = (log x )² and z = e^x .

Then we have to find dy/dz. Now

dy = 2 (logx) 1 = 2 logx
dx               x      x

and    dz/dx = e^x

Thus   dy = dy/dx  = 2 log x)/ x = 2 log x
          dz    dz/dx         e^x             xe^x

(iii)    Let y = x2 + logx +e-x and z = √x

Then    dy = 2x + 1 - e ^-x and dz =  1
           dx           x               dx   2√x

Hence   dv = dv/dx = 2√x . ( 2x + 1 -e ^-x ) .
            dz    dz/dx                     x

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