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.