Find y' if y = x^3/3^x

Question

anonymous · Answer

well there are 5 ways to solve this problem

anonymous · Answer

one of them is by using the quotient rule, or the product rule

anonymous · Answer

$$y =\frac{  x^{3} }{ 3^{x} }$$

anonymous · Answer

this can be rewritten as

anonymous · Answer

$$y = x^{3}. 3^{-x}$$

anonymous · Answer

you know that the product rule is udv +vdu

anonymous · Answer

$$\frac{ dy }{ dx } = 3^{-x}3x^{2} + x^{3}\ln(3)(-3^{-x})$$

anonymous · Answer

$$\frac{ dy }{ dx } = 3^{-x}3x^{2} - x^{3}\ln(3)3^{-x}$$

anonymous · Answer

$$\frac{ dy }{ dx } = x^{2}3^{-x}[3 - x\ln(3)]$$

anonymous · Answer

3^(-x) , not 3^x

anonymous · Answer

use wolfram to get

3*x^2/3^x-x^3*ln(3)/3^x

anonymous · Answer

$$3\,{\frac {{x}^{2}}{{3}^{x}}}-{\frac {{x}^{3}\ln  \left( 3 \right) }{{ 3}^{x}}}$$

anonymous · Answer

well when you take 3^(x) from denominator to numerator it becomes 3^(-x)

anonymous · Answer

and timo used a software and i can see his result is similar to mine

anonymous · Answer

to solve 3^(-x) use d operator differential method

anonymous · Answer

if anyone requires a proof let me know

anonymous · Answer

i will solve ur partial derivative it is simple just give me a moment i am on the phone mr Hoa

anonymous · Answer

don't be upset. we try to help each other, right? if i see your mistake, i just help you to correct it. i, myself need help, too. Sorry if i bother you.

anonymous · Answer

i am not bothered my friend

anonymous · Answer

my answer is correct

anonymous · Answer

100% agree with you that we must solve the problem by hand, because in test, we don't have computer to search the result

anonymous · Answer

$$y = \frac{ x^3 }{ 3^{x} }$$

anonymous · Answer

now what happens if you take 3^x to the numerator

anonymous · Answer

you ask me or the asker?

anonymous · Answer

$$\frac{ 3^{-x} }{ 1 }$$

anonymous · Answer

why? I don't understand, please, explain

anonymous · Answer

well i wanted to simplify the problem by using the product rule and avoiding the quiescent rule

anonymous · Answer

so i took the 3^(x) from the denominator to the numerator so it became 3^(-x)

anonymous · Answer

using basic log and anti log rules

anonymous · Answer

yeah, you are right, because the product must be x^3 * 3^(-x) , your typing lack of (-) . That's it. the whole thing is right