6.   Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

       \frac{1 + \frac{1}{x}}{1- \frac{1}{x}} 

Answers (1)

Given,

f(x)=\frac{1 + \frac{1}{x}}{1- \frac{1}{x}}

Also can be written as 

f(x)=\frac{x+1}{x-1}

Now, As we know the derivative of any function  

\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_2d(\frac{dy_1}{dx})-y_1(\frac{dy_2}{dx})}{y_2^2}

Hence, The derivative of f(x) is 

\frac{d(\frac{x+1}{x-1})}{dx}=\frac{(x-1)d(\frac{d(x+1)}{dx})-(x+1)(\frac{d(x-1)}{dx})}{(x-1)^2}

\frac{d(\frac{x+1}{x-1})}{dx}=\frac{(x-1)1-(x+1)1}{(x-1)^2}

\frac{d(\frac{x+1}{x-1})}{dx}=\frac{x-1-x-1}{(x-1)^2}

\frac{d(\frac{x+1}{x-1})}{dx}=\frac{-2}{(x-1)^2}

Hence Derivative of the function is 

\frac{-2}{(x-1)^2}

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