Question

If `g(x) = x^5(4.5)^x`, what is `g'(x)`?

Answer 1

`(dg)/dx =4.5^x xx x^4xx (5 + 1.50408 xx x)`

Explanation:

First consider `y = a^x`. Applying the log transformation to both sides

`log_e y = x log_e a`

Deriving the equation in both sides we get

`dy/y=dx log_e a`

and finally

`dy/dx = y log_e a=a^x log_e a`

So making now

`g(x) = x^n a^x` we have

`(dg)/dx = d/dx(x^n) xx a^x+x^n xx d/dx(a^x)`

so

`(dg)/dx = n x^{n-1}a^x + x^n a^x log_e a`

substituting `n = 5` and `a = 4.5` we have

`(dg)/dx =5xx 4.5^x x^4 + 1.50408xx 4.5^x x^5`

or

`(dg)/dx =4.5^x xx x^4xx (5 + 1.50408 xx x)`