7.(iii)   For some constants a and b, find the derivative of \frac{x - a }{x -b }

Answers (1)

Given,

f(x)=\frac{x - a }{x -b }

Now As we know the quotient rule of derivative,

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

So applying this rule, we get

\frac{d(\frac{x-a}{x-b})}{dx}=\frac{(x-b)\frac{d(x-a)}{dx}-(x-a)\frac{d(x-b)}{dx}}{(x-b)^2}

\frac{d(\frac{x-a}{x-b})}{dx}=\frac{(x-b)-(x-a)}{(x-b)^2}

\frac{d(\frac{x-a}{x-b})}{dx}=\frac{a-b}{(x-b)^2}

Hence 

f'(x)=\frac{a-b}{(x-b)^2}

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