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amyna
 one year ago
diffrentiate:
g(x)=4^3x^2
Thank you for your help!
amyna
 one year ago
diffrentiate: g(x)=4^3x^2 Thank you for your help!

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triciaal
 one year ago
Best ResponseYou've already chosen the best response.0differentiate power raised to a power think we can do substitution

triciaal
 one year ago
Best ResponseYou've already chosen the best response.0dw:1441505133978:dw

triciaal
 one year ago
Best ResponseYou've already chosen the best response.0maybe the log function

amyna
 one year ago
Best ResponseYou've already chosen the best response.0lol i dont understand. what you wrote, is that the correct way to do this problem or not?

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6Nah. Start by rewriting dw:1441505728153:dw

amyna
 one year ago
Best ResponseYou've already chosen the best response.0oh okay so you have to use logs and rewrite it. is this true for all problems that may look similar to this?

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6Generally base e is easier to deal with. And changing bases only changes the exponent by a constant factor. So it almost always helps.

amyna
 one year ago
Best ResponseYou've already chosen the best response.0wait then how do i solve it after rewriting it? lol i forgot how to do that part!

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6Use the chain rule.

amyna
 one year ago
Best ResponseYou've already chosen the best response.0ok thanks. i think i got it from here

amyna
 one year ago
Best ResponseYou've already chosen the best response.0no i don't get it. i don't know how to solve it. i tried using the chain rule

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6Hm, you might need to review the chain rule then. I can go through this example if you want but I don't think it will be much help in general.

amyna
 one year ago
Best ResponseYou've already chosen the best response.0yes please! i would greatly appreciate that! :)

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[y=(f(x))^{g(x)} , \text{ assume } f(x)>0 \\ \\ \text{ take } \ln( ) \text{ of both sides } \\ \ln(y)=\ln((f(x))^{g(x)}) \\ \ln(y)=g(x) \ln(f(x)) \text{ by use of power rule for logarithms } \\ \\ \text{ now differentiate both sides } \\ \frac{y'}{y}=g'(x) \cdot \ln(f(x))+g(x) \cdot \frac{f'(x)}{f(x)} \\ \text{ by a whole bunch of rules :p } \\ \text{ left hand side I just used chain rule } \\ \text{ right hand side I used product rule and chain rule }\] \[y'=\{g'(x) \ln(f(x))+g(x) \frac{f'(x)}{f(x)} \} y \\ \text{ note: this step I just multiplied both sides by } y \] \[\text{ now remember } y=(f(x))^{g(x)} \\ \text{ so make this replacement and you are done} \\ \]

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.6Ok, I'll try to explain how to apply the chain rule. You might want to look at this lesson from MIT OCW as well: http://ocw.mit.edu/courses/mathematics/1801scsinglevariablecalculusfall2010/1.differentiation/partadefinitionandbasicrules/session11chainrule/ You look at \(e^{(ln4)*3*x^2} \). If it was e^x the derivative would be e^x. But now the exponent is a function of x. So the result is e to that function of x _multiplied by the derivative of that function_. So \([e^{g(x)}]' = g'(x)* e^{g(x)}\) In this case g(x) = (ln4)*3*x^2 g'(x) = (ln4)*6*x so the final result is \[(ln4)*6x*e^{ln4*3*x^2} \] which you can rewrite as \[(ln4)*6x*4^{ln4*3*x^2} \] using the same idea with logarithms that we started with.
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