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mathcalculus
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can someone explain to me Implicit Differentiation:
if y^3 = 25x^2, determine dx/dt when x = 5 and dy/dt = 1
 10 months ago
 10 months ago
mathcalculus Group Title
can someone explain to me Implicit Differentiation: if y^3 = 25x^2, determine dx/dt when x = 5 and dy/dt = 1
 10 months ago
 10 months ago

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mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
these are the steps but i don't understand them :/ 3y^2*dy/dt = 50x*dx/dt When x = 5, y^3 = 25*5^2 y^3 = 625 y = cube root 625 = 8.55 3(8.55)^3*1 = 50*5*dx/dt 1875 = 250dx/dt 7.5 = dx/dt
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
isn't y=2.92402?
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
i did this on the calculator: sqrt (625)^(1/2)
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
(1/3)**
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
@satellite73
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
Let's start with this part... \[\Large \frac{ d }{ dt } y^3\]use the chain rule to differentiate it... so bring down the exponent, reduce the exponent by one, then multiply by the derivative of y w. respect to t \[\Large 3 y^2 *\frac{ dy }{ dt }\]
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
so I can't replace the dy/dt by 1 since it's given?
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
@agent0smith
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
Well yes but i was seeing if you understood the differentiation... don't worry about plugging in numbers till later.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
and 3y^2 is already the derivative, why do we need to use the chain rule?
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
@agent0smith
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
Because you have to multiply by the derivative of y with respect to t.
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
3y^2 is NOT the derivative of y^3, UNLESS you're just differentiating with respect to y.
 10 months ago

tpmys Group TitleBest ResponseYou've already chosen the best response.1
normally we have y dependent on x here y, and x both are dependent on some t, so we dont know what the function is so instead of y^3 >>> 3y^2 we have the chain rule y^3 >>> 3y^2 * y'
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
It's the same with dy/dx... the derivative of y with respect to x is not just 1... it's 1*dy/dx
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
ok... :/ it's a little confusing..
 10 months ago

ranga Group TitleBest ResponseYou've already chosen the best response.0
agent0smith explained it nicely: \[\frac{ d }{ dy }(y ^{3}) =3y ^{2}\] But \[\frac{ d }{ dt }y ^{3} = \frac{ d }{ dy }y ^{3}\frac{ dy }{ dt } = 3y ^{2}\frac{ dy }{ dt }\]
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
It's the same thing with the chain rule \[\huge \frac{ d }{ dx } f(x)^n = n*f(x)^{n1} * f \prime (x)\] you have to always multiply by the derivative at the end.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
why can't we plug in the numbers right away if it's given?
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
You can... you're probably just better off not doing it until you really know what you're doing.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
right i understand that... @ranga
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
well i was taught to plug in.. that's why.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
that's the confusing part.
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
Fair enough... but you can't plug in until you have this step: 3y^2*dy/dt = 50x*dx/dt
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
exactly what i have right now.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
so dy/dt is 1...
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
in the steps above, why is this: 3(8.55)^3*1 = 50*5*dx/dt?
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
y=8.55? y is suppose to be 2.92402.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
and it's raised to the 3rd, and why not 2?
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
When x = 5, y^3 = 25*5^2 y^3 = 625 y = cube root 625 = 8.55 i think the next line after that is a mistake and it should by to the power of 2
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
so y=2.92402 is correct?
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
y^3 = 625 y = cube root 625 = 8.55
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
how?
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
i did this in the calculator: sort(625)^(1/3)
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
sqrt*
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
And you'll get 8.55... 2.94 cubed is close to 3^3 which is 27. Not 625.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
i swear im getting 2.92402
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
i plugged it in exactly like iwrote it.
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
sort(625)^(1/3) why are you taking the square root of a cubed root...
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
y^3 = 625 y = 625^(1/3)
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
OHHH
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
i thought the sqrt could do cube root.
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
if we did the exponent also.
 10 months ago

agent0smith Group TitleBest ResponseYou've already chosen the best response.3
That would mean ((625^(1/3))^(1/2) = 625^(1/6)
 10 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
thank you! i got it !
 10 months ago
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