Thread: How to diffrentiate an exponential function to the function base ?

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

Originally Posted by Nora

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

Consider that

Now use the chain rule to get the derivative of that last term.

Thank you could you please tell me why use e^ln(y) ?(i mean the last two steps )

Originally Posted by Nora

Thank you could you please tell me why use e^ln(y) ?(i mean the last two steps )

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:

aha I got it .. So it's a rule like other rules (Exponential , product , log ....from where you can get directly the derivative

Thank you so much

I'll solve the equation right now

Originally Posted by Nora

aha I got it .. So it's a rule like other rules (Exponential , product , log ....from where you can get directly the derivative

Thank you so much

I'll solve the equation right now

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

Hello, Nora!

Use logarithmic differentiation.

Take logs: .

Differentiate implicitly: .

. . .

. .

. .

i was looking for this problem

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
.

Originally Posted by HallsofIvy

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
.

excellent

Thank you very much for helping me
Soroban

HallsofIvy