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Take the first derivative using the chain rule and then take the derivative of that.
Second derivative will require both the chain rule and the product rule.
could you show me it worked step by step
That's the first derivative. Do you see where things might have come from?
yes i had it all right except the 2x and thats by taking the derivative of the things inside the brackets
Yes :) that is the chain rule.
So now we move on to the second derivative which involves the product rule.
Do you know how to use the product rule?
okay i got.... 4(x^2+9)^3 (2) + (2x) 12(x^2+9)^2
Very close. First half is correct. Second half you forgot something.
what did i forget
oh do the product rule to finish?
the books has 8(x^2+9)^3 + 48x^2(x^2+9)^2 = 8(x^2+9)^2(7x^2+9)
Well the book is correct lol. All you forgot in your was the chain rule on the second half. Should have been (2x)12(x^2+9)(2x) that 2x on the end is what you forgot.
why is the 2x at the end as well?
Ok so when you did the second half of the product rule you can say to yourself (2x) times the derivative of the function. Well the derivative of that function 4(x^2+9)^3 is 12(x^2+9)^(2) times the chain rule (2x).
okay so now where do they get their answer
got the first part, but didnt get it after the simplifying
They probably multiplied stuff out and then it eventually got them to that answer. Truthfully, I have never seen a teacher who would not accept the first answer you have. The other answer is just tedious work.
Actually I just looked at their second answer and you could factor...
okay could you help me with a ln y''?
okay. I am horrible at ln problems but here is the original y'' for y= ln x/x^2
Ok this is a quotient rule problem or can be a product rule if you rearrange it. Take your pick.
SO IT COULD BE (LN X)(-X^2)
Well could be lnx(x^-2). Normally a lot of people prefer to use the product rule when possible, but it's a little easier.
what is ln x '
The derivative of lnx is 1/x
oh yeah sorry
No worries :)
okay so y' is lnx(-2x^-3) + (x^-2)(1/x)
okay now im going to need some help..lol
no just straight so i can see what to do
I'm confused. So what do you need help on? Sorry lol
Ah. Ok. Well you can use the product rule again for the second derivative, but twice. So use the product rule for the lnx(-2x^-3) and then add that with the result of the product rule of (x^-2)(x^-1). I rewrote 1/x so it's easier to see.
can you show me it worked out?