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

4440 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

g(x) = x/(x+3) + 3(2x+1)/(x^2 +x - 6) a)Show that g(x) =(x+1)/(x-2), x>3 b)Find the range of g c)Find the exact value of a for which g(a)=g^(-1)(a).


Differentiate f(x)= x^3 + x^(1/3)-2


How do I differentiate (x^2 + 3x + 3)/(x+3)


Find dy/dx for (x^2)(y^3) + ln(x^y) = 5sin(6x)/x^(1/2)


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