{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

3.7 Implicit Differentiation

# 3.7 Implicit Differentiation - (3.7 Implicit Dierentiation...

This preview shows pages 1–2. Sign up to view the full content.

(3.7) Implicit Differentiation. Example. Find d dx y 3 if y = cos x . We get - 3 cos 2 x sin x by the chain rule. What if y = x 4 + 5 x ? Then we get 3( x 4 + 5 x ) 2 (4 x 3 + 5). In each case, we get 3 y 2 dy dx . This makes sense by the chain rule: Say w = y 3 . Then dw dx = dw dy dy dx = 3 y 2 dy dx . Example. Now try 1. x 2 y 3 2. x 2 /y 4 3. sin y Implicit functions. We can specify a function y explicitly, by stating a formula for it. Say y = 25 - x 2 . (‘Explicit’ means ‘fully revealed’.) But we can specify the same function implicitly: x 2 + y 2 = 25. We understand that we can solve for y to obtain the explicit formula. (This also gives y = - 25 - x 2 implicitly.) ‘Implicit’ means ‘capable of being understood from something else though unexpressed’. Another example: The function given implicitly by y 3 +3 y = 2 x can be given explicitly by solving for y . However, this is not easy in this case. It turns out y = 3 x + x 2 + 1 + 3 x - x 2 + 1 . It often happens that it is difficult or impossible to deal with a function explicitly, and instead, we must deal with it implicitly. (This is particularly true in the subject known

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern