For example ,
y = 17(22+x)^(41-x)^30
I tried to use math code but it didn't show the exponential numbers/function properly
Why? - Well, because you do not have a rule to differentiate something like directly (analogous to the product rule or the quotient rule, that is). Instead you do it like this:
where the last equality holds because of the chain rule. Now you can apply the product rule (and for the chain rule again) and are done:
Um, well, yes you might take it as a rule like the other rules, but you would not want to actually memorize it, like you did (or should do) with the other, much more important rules: first, because it is not used that often, and, second, because it is easier to remember that you can replace by and then apply the rules that you already know about.
P.S: Please note that I have made a silly mistake in my first reply to your question that I have now corrected: the constant factor 17 in front of the power-term can (and should) be left as a factor in front of the remaining exponential mess...
well , It's already written in my book .. when you've said it's a rule I grabbed my book to look for it then I found it under (exponential function to the base of constant ) and compare it to your explanation it was so helpful
Thank you
and yes the number 17 i knew it was mistake because it's constant and can't be included in derivative thank you for clarifying
you made my calculus life easier
With something like " ", there are two simple mistakes we could make:
1) Treat the exponent, g(x), as a constant and use the "power rule" to get
" ".
2) Treat the base, f(x), as a constant and use the "exponential rule" to get " .
The interesting thing is that the correct derivative is the sum of those two errors!
Taking the logarithm of both sides of , . On the left side, . On the right, using the product rule, .
Multiplying on both sides of by , we get
.