y = 10^(cot(x))To differentiate it by the chain rule, substitute u = cot(x)

y` = d/du 10^u * d/dx uNow 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 uNow 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 uSubstituting 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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