## Derivative of A Function With Respect To Another F

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# Derivative of A Function With Respect To Another Function

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

dz dz/dx g '(x)

Example . Differentiate

(i) (logx)

(ii) x

Then we have to find dy/dz. Now

dx x x

and dz/dx = e

Thus

dz dz/dx e

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

Then

dx x dx 2√x

Hence

dz dz/dx x

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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) x

^{2}+ 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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