Basic Question: Differentiating Related Functions (Calculus)

[Adeste]

EXAMPLE 1
Find dx/dt:

xy = 4

To solve, you start with:

x • dy/dt + y • dx/dt = 0

EXAMPLE 2

Find dy/dt:

12x² = y³ - 24

In contrast to Example 1, you begin solving the problem with x • dx/dt:

24x • dx/dt = 3y² • dy/dt

As you can see, one solution begins with x • dy/dt and another solution begins with x • dx/dt.

Why?

COMPLETE WORKED PROBLEMS (FOR REFERENCE):

EXAMPLE 1:

Given:

xy=4

dy/dt=1

y=.5

Find dx/dt

Solution to Example 1:

x = 4/y = 4/.5 = 8

x • dy/dy + y • dx/dt

8•1 + .5 dy/dt = 0

dx/dt = 8/.5=16

EXAMPLE 2:

Given:

12x² = y³ - 24

dx/dt= -6

x= -.5

Find dy/dt

Solution to Example 2:

12•(-.5)² = y³ - 24

27=y³

y=3

24x • dx/dt = 3y² * dy/dt

24•(-.5)²•-6 = 3•(3)² •dy/dt

dy/dx = 72/27 = 8/3

 

[Adeste]

EXAMPLE 1



Find dx/dt:

xy = 4

To solve, you start with:

x • dy/dt + y • dx/dt = 0



EXAMPLE 2



Find dy/dt:

12x² = y³ - 24

In contrast to Example 1, you begin solving the problem with x • dx/dt:

24x • dx/dt = 3y² • dy/dt

As you can see, one solution begins with x • dy/dt and another solution begins with x • dx/dt.



Why?



COMPLETE WORKED PROBLEMS (FOR REFERENCE):



EXAMPLE 1:



Given:

xy=4

dy/dt=1

y=.5

Find dx/dt



Solution to Example 1:



x = 4/y = 4/.5 = 8



x • dy/dy + y • dx/dt



8•1 + .5 dy/dt = 0



dx/dt = 8/.5=16



EXAMPLE 2:



Given:

12x² = y³ - 24

dx/dt= -6

x= -.5

Find dy/dt



Solution to Example 2:



12•(-.5)² = y³ - 24

27=y³

y=3



24x • dx/dt = 3y² * dy/dt



24•(-.5)²•-6 = 3•(3)² •dy/dt



dy/dx = 72/27 = 8/3

 

[Dr.Peterson]

As you can see, one solution begins with x • dy/dt and another solution begins with x • dx/dt.

Why?

They both really start the same way: implicit differentiation. You are differentiating both sides with respect to t.

The only difference is that the different forms of expressions cause you to use different methods, in the first case the product rule, and in the second the power rule (both in combination with the chain rule).

The derivative of xy is x dy/dt + y dx/dt; the derivative of 12x² is 24x dx/dt.

 

[Steven G]

EXAMPLE 1
Find dx/dt:

xy = 4

To solve, you start with:

x • dy/dt + y • dx/dt = 0

Addition does commute so you can have dx/dt first. That is y • dx/dt +x • dy/dt = 0
I am not sure at all why you want dx/dt 1st but as I showed it is easily obtained.

 

[Adeste]

Thanks! I tried asking a lot of people and you're the only one that got to the heart of the matter. I get people think it's a dumb question, but sometimes that's just where you are ;-)