Calculus: Derivatives of n^x treatment?

Question

In Calculus, are any number to the power of x treated as E^x when working with derivatives? (i.e. 10^x ... 10xln(10)

Answer 1

Use the limit definition of derivatives to find the derivative of the expressions

 

e^x   ,     C^x     ,    and C^xlnC

 

 

If you get the same limit, then any number to the power of x can be treated as e^x.

Answer 2

If n is positive, then the derivative of y = n^x can be found by logarithmic differentiation.  

 

y = n^x

 

lny = ln(n^x)

 

lny = xln(n)

 

y'/y = ln(n)

 

y' = yln(n) 

 

y' = ln(n)(n^x)