Find dy/dx.
y=(3x^4+4x^3-1)/x^3

Question

Find dy/dx.
y=(3x^4+4x^3-1)/x^3

anonymous · Answer

$$y = \frac{ 3x^4+4x^3-1 }{ x^3 }$$

anonymous · Answer

So what rule would you use? :)

anonymous · Answer

quotient rule, right?

psymon · Answer

Break it into 3 fractions o.o

psymon · Answer

$$\frac{ 3x^{4}+4x^{3}-1 }{ x^{3} }= \frac{ 3x^{4} }{ x^{3} }+\frac{ 4x^{3} }{ x^{3} }-\frac{ 1 }{ x^{3} }$$

anonymous · Answer

originally, I got 3x^6+3x^2/x^5 but I don't know how to fix it to make it correct

psymon · Answer

Okay, well you can do it with the 3 fractions, because that ends up making it much easier. But is that whatyou got trying to do quotient rule?

anonymous · Answer

yeah, that's what I was taught but i did something wrong

psymon · Answer

Well, I mean, of course you cna quotient rule, I just mean to say that there's an easier way with this problem that would be nice to try to recognize. But okay, lets say we HAD to do it with quotient rule or get an F or something:

$$\frac{ f'(x)g(x) - f(x)g'(x) }{ [g(x)]^{2} }$$ SO f is the top function and g is the bottom function. Ill just stack the information and then combine it in later:

f(x) = 3x^4 +4x^3 - 1
g(x) = x^3
f'(x) = 12x^3 + 12x^2
g'(x) = 3x^2

$$\frac{ (12x^{3}+12x^{2})(x^{3}) - (3x^{4}+4x^{3}-1)(3x^{2}) }{ x^{6} }$$
$$\frac{ 12x^{6}+12x^{5}-(9x^{6}+12x^{5}-3x^{2}) }{ x^{6} }$$
$$\frac{ 12x^{6} + 12x^{5} - 9x^{6} - 12x^{5} + 3x^{2} }{ x^{6} }$$
$$\frac{ 3x^{6}+3x^{2} }{ x^{6} } = \frac{ 3x^{2}(x^{4}+1) }{ x^{6} }\implies \frac{ 3(x^{4}+1) }{ x^{4} }$$

And you can separate that into two fractions if you wanted,but yeah.

anonymous · Answer

okay, i got as far as the last line there until it simplified into the final answer. how did you get x^4 in the denominator?

psymon · Answer

Can you see how I factored it into 3x^2(x^4+1) on top? do you understand up to that point or is it further back?

anonymous · Answer

I understand right up to that point

anonymous · Answer

ohhh do i subract the exponents? is that how the x^4 got in the denominator?

psymon · Answer

Well, I just factor the 3x^6 + 3x^2 portion. In the same way I could factor something like
3x+9 into 3(x+3), I just did that with the top. That allows me to pull out a greatest factor of 3x^2 to give me 3x^2(x^4+1). 

And yes, exactly. Or you could even think of it like this:

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