Question

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)`

Answer 1

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)`