Finding the Derivative When the Exponent is a Polynomial & a Chain Rule Question

Question

I was blessed with the equation of 597.3(0.921^4x+12). I've tried multiple ways to find the derivative, but I keep coming to a couple road blocks.

I'm certain that I have to use the chain rule to figure it out, but wouldn't the derivative of 597.3 with nothing on it just come to 0? Then that would eventually make the answer to the whole question 0 too, so I'm sure that I'm missing something.

On the other hand, I have no clue how to find the derivative of 0.921^4x+12 since I can't find anywhere online that'll tell me what to do when the exponent is a polynomial. All the results were just if the equation was a polynomial, & this was a little past that level.

Any help is really appreciated

Answer 1

Chain rule is correct. The 597.3 will not disappear because it is multiplying the function, not a constant that is added to it. This function is an exponential function, the derivative of which is an exponential with the same base. When e is the base, it is its own derivative. In this case, because the base isn't e, we use this rule:

d/dx [a·b^x] = a·lnb·b^x so [597.3·.921^x]^' = 597.3·ln(.921)·.921^x

But then we need to apply chain rule to the exponent function, so we get 4·597.3·ln(.921)·.921^4x+12