How do I differentiate implicitly?

The most important thing to remember when differentiating implicitly is that y is a function of x. Rewriting y as y(x) often makes it much clearer. For example, evaluate d/dx (y2): using the aforementioned notation, this becomes d/dx [y(x)]2. By the chain rule, it is easy to see that this is equal to dy/dx * 2y.

Perhaps an easier way of remembering this is to differentiate with respect to y, then multiply by dy/dx. For example, evaluate d/dx(ln(y)): to find the answer, we differentiate ln(y) with respect to y to get 1/y, then multiply this by dy/dx to get dy/dx * 1/y.

The above method works because of the chain rule, which states that df/dx = df/dy * dy/dx. All we are doing is renaming the function as f (in the first example f = y2, in the second example f = ln(y)) and applying this result.

SG
Answered by Seb G. Maths tutor

4391 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

What is the gradient of y = xcos(x) at x=0?


Differentiaate the folowing equation with respect to x: y=4x^3-3x^2+9x+2


Let y=arcsin(x-1), 0<=x<=2 (where <= means less than or equal to). Find x in terms of y, and show that dx/dy=cos(y).


Express 9^(3x + 1) in the form 3^y , giving y in the form ax + b, where a and b are constants.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences