Find the derivative of cot log(x)

We are asked to find the derivative of cot(log(x)):
(1) Assuming that log(x) is the common logarithm (logarithm base 10) of x:
Note that the derivative of cot(u) (where u is a differentiable function of x) is given by -csc^2(u) du .
Also, d/(dx) log(x)=1/(xln(10)) where ln(10) is the natural logarithm (base e) of 10.
So using the chain rule we get: d/(dx)cot(log(x))=-csc^2(log(x))*1/(xln(10))=-(csc^2(ln(x)/ln(10)))/(xln(10))
(2) if by log(x) you meant the natural logarithm of x ( or log base e of x) then:
d/(dx)cot(log(x))=-csc^2(log(x))*1/x=-(csc^2(log(x)))/x
In both cases we have the derivative of the cotangent, which is the negative of cosecant squared, of the function logarithm of x. This is multiplied by the derivative of the logarithm function. In the case of the natural logarithm the derivative is just 1/x, while in the case of the natural logarithm we have a constant factor equal to the natural logarithm of 10.
http://mathworld.wolfram.com/Cotangent.html