Determine y' for xy = 1 by implicit differentiation, Mathematics

Assignment Help:

Determine y′ for xy = 1 .

Solution : There are in fact two solution methods for this problem.

Solution 1: It is the simple way of doing the problem.  Just solve for y to obtain the function in the form which we're utilized to dealing with and then differentiate.

y = 1/x ⇒             y′ = - 1/x2

Hence, that's easy sufficient to do.  However, there are some functions for which it can't be done. That's where second solution method comes to play.

Solution 2 (through implicit differentiation):

In this we're going to leave the function in the form which we were given & work with it in that form.  Though, let's recall from the first part of this solution that if we could solve out for y then we will get y like a function of x.  In other terms, if we could solve out for y (as we could in this case, however won't always be capable to do) we get y = y (x).  Let's rewrite the equation to note down this.

                                          xy = x y ( x ) = 1

Be careful here and note down that while we write y ( x ) we don't mean y times x.  What we are noting at this time is that y is some (probably unknown) function of x. It is important to recall while doing this solution technique.

In this solution the next step is to differentiate both sides w.r.t. x as follows,

                                 d ( x y ( x ))/ dx = d (1)/ dx

The right side is simple.  It's just the derivative of constant. The left side is also simple, but we've got to identify that we've in fact got a product here, the x and they ( x ) .  Thus to do the derivative of the left side we'll have to do the product rule.  By doing this gives,

 (1) y ( x ) + x d ( y ( x )) /dx= 0

Now, recall that we have the given notational way of writing the derivative.

d ( y ( x )) / dx = dy/ dx = y′

By using this we get the following,

y + xy′ = 0

Note as well that we dropped the ( x ) on the y as it was just there to remind us that the y was a function of x & now that we've taken the derivative it's no longer needed really. We just desired it in the equation to identify the product rule while we took the derivative.

thus, let's now recall just what were we after. We were after the derivative,  y′ , and notice that there is now a  y′ in the equation.  Thus, to get the derivative all that we have to do is solve the equation for  y′ .

                                                                   y′ = - y/ x

There it is. By using the second solution technique it is our answer. It is not similar with the first solution however. Or at least it doesn't look like the similar derivative that we got from the first solution.  However, recall that we actually do know what y is in terms of x and if we plug that in we will get,

                                            y′ = -       (1/x) /x= -1/ x2

that is what we got from the first solution.  Regardless of the solution technique utilized we should get the same derivative.


Related Discussions:- Determine y' for xy = 1 by implicit differentiation

Shares and divident, A man invest ?13500 partly in shares paying 6% at ?140...

A man invest ?13500 partly in shares paying 6% at ?140 and partly in 5% at 125.If he is tolal income is 560, how much has he invested in each?

How to calculate probability of event, Q. How to calculate Probability of e...

Q. How to calculate Probability of event? Ans. What chance do I have to toss the coin and get a head? You might think 50-50, 50%. What about tossing it 5 times and getting

Proof of: limq?0 (cosq -1)/q = 0 trig limit, Proof of: lim q →0 (co...

Proof of: lim q →0 (cos q -1) / q = 0 We will begin by doing the following, lim q →0 (cosq -1)/q = lim q →0 ((cosq - 1)(cosq + 1))/(q (cosq + 1)) = lim q

Reduction-types of word problems related to subtraction, Reduction -when t...

Reduction -when the original amount and the balance or remainder are known, to find the part that has been given away. (e.g., there were 15 toffees in a container, and there are

How to multiplying rational expressions, how to Multiplying Rational Expres...

how to Multiplying Rational Expressions ? To multiply fractions, or rational expressions, you must multiply the numerators and then multiply the denominators. Here's how it is

Find x if one ball is drawn at random from the box, A box contains 12 balls...

A box contains 12 balls out of which x are black .if one ball is drawn at random from the box  what is the probability that it will be a black ball ? If 6 more black balls are   pu

Law of cosines - vector, Theorem a → • b → = ||a → || ||b → || cos• ...

Theorem a → • b → = ||a → || ||b → || cos• Proof Let us give a modified version of the diagram above. The three vectors above make the triangle AOB and note tha

Geometria, un prisma retto ha per base un rombo avente una diagonale lunga ...

un prisma retto ha per base un rombo avente una diagonale lunga 24cm. sapendo che la superficie laterale e quella totale misurano rispettivamente 2800cm e3568cm ,calcola la misura

Calculate the throughput and link utilization, 4. Two hosts, one on East (h...

4. Two hosts, one on East (host A) and one on the west coast (host B) of the USA are exchanging data. Suppose A is sending a large file to B. The file is split into packets of size

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