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What is the derivative of `log_(x)5` with respect to `log_(5)x`?A. `-(log_(5)x)^(-2)`B. `(log_(5)x)^(-2)`C. `-(log_(x)5)^(-2)`D. `(log_(x)5)^(-2)` |
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Answer» Correct Answer - A Let `u_(1)=log_(x)5 and u_(2)=log_(5)x` `rArr u_(1)=(log_(e)5)/(log_(e)x) and u_(2)=(log_(e)x)/(log_(e)5)` On differentiating w.r.t. x, we get `(du_(1))/(dx)[(log_(e)x(0)-((1)/(x)))/((log_(e)x)^(2))]log_(e)5=-(log_(e)5)/(x(log_(e)x))` `and (du_(2))/(dx)=(1)/(xlog_(e)5)` `therefore(du_(1))/(du_(2))=(du_(1)//dx)/(du_(2)//dx)=-(log_(e)5)/(x(log_(e)x)^(2))xx x log_(e)5` `=-((log_(e)5)/(log_(e)x))^(2)=-(log_(x)5)^(2)=-(log_(5)x)^(-2)` |
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