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JEE Main & Advanced Mathematics Differentiation Question Bank Derivative at a point Standard Differentiation

  • question_answer
    If \[f(x)\] is a differentiable function, then \[\underset{x\to a}{\mathop{\lim }}\,\frac{af(x)-xf(a)}{x-a}\] is                    [UPSEAT 2002]

    A)            \[a{f}'\,(a)-f\,(a)\]

    B)            \[af\,(a)-f'(a)\]

    C)            \[a{f}'\,(a)+f\,(a)\]

    D)            \[af\,(a)+f'(a)\]

    Correct Answer: A

    Solution :

               \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{af(x)-xf(a)}{x-a}\]Þ  \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{af(x)-xf(a)+af(a)-af(a)}{x-a}\]            Þ  \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{a[f(x)-f(a)]-f(a)[x-a]}{x-a}\]            Þ  \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{a[f(x)-f(a)]}{x-a}-\underset{x\to a}{\mathop{\lim }}\,\,f(a)\] Þ \[a{f}'(a)-f(a)\].


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