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

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mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0these 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

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0i did this on the calculator: sqrt (625)^(1/2)

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3Let'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 \[\Large 3 y^2 *\frac{ dy }{ dt }\]

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0so I can't replace the dy/dt by 1 since it's given?

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

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

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3Because you have to multiply by the derivative of y with respect to t.

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.33y^2 is NOT the derivative of y^3, UNLESS you're just differentiating with respect to y.

tpmys
 one year ago
Best ResponseYou've already chosen the best response.1normally 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'

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

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0ok... :/ it's a little confusing..

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

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3It'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.

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0why can't we plug in the numbers right away if it's given?

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3You can... you're probably just better off not doing it until you really know what you're doing.

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0right i understand that... @ranga

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0well i was taught to plug in.. that's why.

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0that's the confusing part.

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

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0exactly what i have right now.

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

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0y=8.55? y is suppose to be 2.92402.

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0and it's raised to the 3rd, and why not 2?

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3When 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

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0so y=2.92402 is correct?

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3y^3 = 625 y = cube root 625 = 8.55

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0i did this in the calculator: sort(625)^(1/3)

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3And you'll get 8.55... 2.94 cubed is close to 3^3 which is 27. Not 625.

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0i swear im getting 2.92402

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0i plugged it in exactly like iwrote it.

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

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3y^3 = 625 y = 625^(1/3)

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0i thought the sqrt could do cube root.

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0if we did the exponent also.

agent0smith
 one year ago
Best ResponseYou've already chosen the best response.3That would mean ((625^(1/3))^(1/2) = 625^(1/6)

mathcalculus
 one year ago
Best ResponseYou've already chosen the best response.0thank you! i got it !
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