1. I suggest you post the entire question, as there is no way I can see atm how can become without the context...
2. I can't tell if your original function is or . Please use brackets where they're needed to avoid ambiguity.
Okay, guys, I have two questions this time:
1. If then how does it become ? Shouldn't it become ? My book did the in one step, before differentiating. The solution really threw me off.
2. Is the derivative of or ? Is the a separate term and thus differentiated separately? If that is the case, would I use the same convention to deal with ?
Thanks in advance guys for all the help.
For 1. the book just simply says "rewrite the function as before they begin to differentiate. There was no step in between to explain it, which threw me off.
For 2. I was referring to , so would it be differentiated as two terms or did they use the product rule to differentiate? Also, how would differentiating and even be different?
1. The original function is , so would you say the book probably made an error and differentiated for a different function?
As for 2, to apply the chain rule don't both terms need to be in the brackets? So, wouldn't it be the product rule to differentiate ? I am not saying you are wrong, but I seem to have the most trouble with this type of function.
Also, how would you differentiate and ? For the former do we use the chain rule and thus differentiate as ? And for would it become
?
Thanks in advance for all the help.
I'm not sure, I was unsure before as to what the original function was.
Do you understand why we can rewrite:
Hereafter you can use the chain rule to give .
No, you have a function (x^2) within another function cos(x^2) so the chain rule applies. The product rule applies if you have two functions multiplied together. For example if then you'd need the product rule.As for 2, to apply the chain rule don't both terms need to be in the brackets? So, wouldn't it be the product rule to differentiate ? I am not saying you are wrong, but I seem to have the most trouble with this type of function.
The same as above, using the chain rule. If possible avoid this notation as it's unclear.Also, how would you differentiate
It may help you visualise . You can then use the chain rule. I'm not sure where your 0 power comes from though.and ? For the former do we use the chain rule and thus differentiate as ? And for would it become
?
Thanks in advance for all the help.
If you use the power rule as normal (ignoring the cos(x) for now and leaving it there) we get . However, we're not done since we need to multiply by the derivative of cos(x) [which we ignored previously] which is
Multiplied this is . If you know your double angle identities you may also write is as
Thanks, e^(i*pi). That cleared up everything for me. One last question: whenever you have a power inside a bracket and outside the bracket, you can multiply them out like you just did so it can be easier to simplify, correct? Also, shouldn't it it become Instead of ?