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Power rule and exponential rule of derivatives

Sometimes it can be confusing whether to use the power rule or the exponential rule of derivatives. For example, consider the following function:
y = 10^(cot(x))
To differentiate it by the chain rule, substitute u = cot(x)
y` = d/du 10^u * d/dx u
Now will you use the power rule or exponential rule on 10^u ?

The power rule is used when
  • The exponent is a number
  • The base is a variable or algebraic expression
The exponential rule is used when
  • The exponent is a variable, or algebraic expression
  • The base is a number
In 10^(cot(x)), the base is a number 10 and the exponent is an expression cot(x), so applying the exponential  rule on it is correct. Then the derivative is:
y` = 10^u * ln(u) * d/dx u
Now substitute back u = cot(x),
y` = 10^(cot(x)) * ln(cot(x)) * d/dx cot(x)
Derivative of cot(x) is -csc^2(x)
y` = 10^(cot(x)) * ln(cot(x)) * -csc^2(x)
On the other hand, if you used the power rule on it, the derivative of 10^u would be u*10^(u - 1)
 y` = u * 10^(u - 1) * d/dx u
Substituting u = cot(x),
y` =  cot(x) * 10^( cot(x) - 1) * d/dx cot(x)
Derivative of cot(x) is -csc^2(x)
y` =  cot(x) * 10^( cot(x) - 1) * -csc^2(x)
... which is a completely different derivative than the one when exponential rule is used.

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