HOW TO DERIVATE FUNCTIONS POWERS WHERE THE BASE AND THE EXPONENT ARE FUNCTIONS?

Here we are in the presence of a composite function, where the most external function is Xⁿ
 
and the innermost is X, but in this case the exponent n is also a function, so the general form of these functions is: Y=(f(x))^g(x)  This type of functions can be solved by applying logarithmic derivation, but there is a general formula to derive them which is:

Y' = g'(x)(f(x))^g(x)Lnf(x) + g(x)f'(x)(f(x))^g(x)-1

To facilitate the calculations, only the two functions that make up Y must be identified and their derivatives calculated, and then carefully included in the formula. Let's see 5 examples.

Calculate Y'

1)    Y= z^3z   The base function is : f(z)= z and the  exponent function is: g(z)= 3z  

 Y' = 3(z)^3zLn(z) + 3z(1)(z)^3z-1

2)    Y= (sent)^cost  The base function is : f(t)= sent and the  exponent function is: g(t)= cost

  Y' = -sent(sent)^costLn(sent) + cost(cost)(sent)^cost-1

3)    Y= (tangt + cost)^t+3  The base function is : f(t)= (tgt + cost) and the  exponent function is:g(t)= t+3

  Y' = t(tangt + cost)^t+3Ln(tangt + cost) + t+3(sec²t -sent)(tgt + cost)^t+2

4)    Y= (secx)^cotgx  The base function is : f(x)= secx and the  exponent function is: g(x)= cotgx

Y' = -cosec²x(secx)^cotgxLn(secx) + cotgx(tgx.secx)(secx)^cotgx-1

5)    Y= (Lnx)^cosecx  The base function is : f(x)= Lnx and the  exponent function is: g(x)= cosecx

  Y' = -cotgx.cosecx(Lnx)^cosecxLn(Lnx) + cosecx(1/x)(Lnx)^cosecx-1

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