Example of implicit differentiation, Mathematics

Assignment Help:

Example of Implicit differentiation

So, now it's time to do our first problem where implicit differentiation is required, unlike the first example where we could actually avoid implicit differentiation by solving for y.

Example   Determine y′ for the following function.

                                                   x2 + y 2  = 9

Solution

Now, it is just a circle and we can solve out for y which would give,

1797_implicite derivation.png

Prior to starting this problem we stated that we must do implicit differentiation here since we couldn't just solve out for y and still that's what we just did.  Thus, why can't we utilize "normal" differentiation here? The problem is the " ±".  With this in the "solution" for y we illustrates that y is actually two different functions. Which should we use?  Should we utilize both? We just want a single function for the derivative and at best we contain two functions here.

Thus, in this example really we are going to have to do implicit differentiation thus we can ignore this. In this instance we'll do the similar thing we did in the first example & remind ourselves that y is actually a function of x and write y as y (x) .  Once we've done it all we have to do is differentiate each term w.r.t x.

                                           dx2 [y ( x )]2  / dx = d (9)/dx

As with the first example the right side is simple.  The left side is also pretty simple as all we have to do is take the derivative of each of term and note  as well that the second term will be same the part (a) of the second example.  All we have to do for the second term is utilizes the chain rule.

After taking the derivative we contain,

                           2 x + 2 [y ( x ) ]1y′ ( x ) = 0

 At this instance we can drop the ( x ) part since it was only in the problem to help with the differentiation procedure. The last step is to just solve the resulting equation for y′ .

2x + 2 yy′ = 0

y′ = - x /y

We can't just plug in for y as we wouldn't know which of the two functions to utilization.  Most answers from implicit differentiation will include both x & y so don't get excited regarding that when it happens.


Related Discussions:- Example of implicit differentiation

Area in polar cordinates, find the area of the region within the cardioid r...

find the area of the region within the cardioid r=1-cos

Forced - damped vibrations, It is the full blown case where we consider eve...

It is the full blown case where we consider every final possible force which can act on the system. The differential equation in this case, Mu'' + γu'  + ku = F( t) The displ

Managment Science, Classify models based on the degree of their abstraction...

Classify models based on the degree of their abstraction, and provide some examples of such models.

Surface areas and volumes, a conical vessel of radius 6cm and height 8cm is...

a conical vessel of radius 6cm and height 8cm is completely filled with water.a sphere is lowered into the water and its size is such that when it touches the size it is immersed.w

Basics of series - sequences and series, Series - The Basics That top...

Series - The Basics That topic is infinite series.  So just define what is an infinite series?  Well, let's start with a sequence {a n } ∞ n=1 (note the n=1 is for convenie

Solve the second order differential equations, Solve the subsequent IVP ...

Solve the subsequent IVP Y'' - 9 y = 0, y(0) = 2, y'(0) = -1 Solution First, the two functions  y (t ) = e 3t  and  y(t ) = e -3t That is "nice enough" for us to

Determine series is convergent or divergent by root test, Find out if the f...

Find out if the following series is convergent or divergent. Solution There really is not very much to these problems another than calculating the limit and then usin

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd