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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
 9 months ago
 9 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
 9 months ago
 9 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
 9 months ago

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

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

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

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
@satellite73
 9 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 \[\Large 3 y^2 *\frac{ dy }{ dt }\]
 9 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?
 9 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
@agent0smith
 9 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.
 9 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?
 9 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
@agent0smith
 9 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.
 9 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.
 9 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'
 9 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
 9 months ago

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
ok... :/ it's a little confusing..
 9 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 \[\frac{ d }{ dt }y ^{3} = \frac{ d }{ dy }y ^{3}\frac{ dy }{ dt } = 3y ^{2}\frac{ dy }{ dt }\]
 9 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.
 9 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?
 9 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.
 9 months ago

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

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

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
that's the confusing part.
 9 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
 9 months ago

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

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
so dy/dt is 1...
 9 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?
 9 months ago

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

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
and it's raised to the 3rd, and why not 2?
 9 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
 9 months ago

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

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

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

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
sqrt*
 9 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.
 9 months ago

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

mathcalculus Group TitleBest ResponseYou've already chosen the best response.0
i plugged it in exactly like iwrote it.
 9 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...
 9 months ago

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

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

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

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

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