what is implicit differentiation and how is it achieved?

First off, an explicit definition needs to be understood. An explicit definition is the standard form in which many equations are found:

y = f(x)        (1)

for example:  y=x2+2

Here, all values of y can be explicitly defined by a function of x. An implicit expression is one where the equation cannot easily be rearranged/not be rearranged into a form similar to the one found in (1) where y can be expressed as a function of x.

In these situations the expression is said to be implicitly defined.

Implicit differentiation is the method in which these equations can be correctly and easily differentiated.


in implicit differentiation, y terms are considered to be a function of a function, or more simply:


y => f(y)


thus when differentiating a y term, the chain rule (or the rule for differentiating a function of a function) needs to be used.

dy/dx=dy/du * du/dx                (2)


also when differentiating y in terms of x, or 

d/dx(f(y))=d/x(y) = dy/dx         (3)


so for example lets look at the following equation:

16y3 + 9x2y - 54x = 0           (4)


lets focus on the first term in this equation to start with 16y3

let A = 16y3

d/dx(A)=d/dx(16y3)                  (5)

here we have to use the chain rule. 

here u=y, thus looking back at (2) our terms are:

dy/du = d/du(16u3) = 48u2         (6)

du/dx = d/dx(y)  = dy/dx            (7)

thus combining (6) and (7) to get the answer for (5) you get:

d/dx(A) = 48u* dy/dx = dy/dx(48y2)            (8)

as you can see from (8), the desired term dy/dx is generated but the y term is still present, this is because y cannot be explicitly defined and this the gradient of the line is dependent on both x and y terms.

following this through for the next two terms in (4) you get:


for 9x2y, product rule is required. note that as this term includes an x term, these terms are differentiated normally.

d/dx(B) = d/dx(9x2y) = d/dx(9x2)*y + d/dx(y​)*9x= 18xy + dy/dx(9x2)


for -54x, this is simple differentiation:

d/dx(C) = d/dx(-54x) = -54


collecting the 3 differentiated terms of A,B and C, you get the answer to:

d/dx(16y3 + 9x2y - 54x) =  d/dx(A) + d/dx(B) + d/dx(C) = dy/dx(48y2) + 18xy + dy/dx(9x2) - 54 = 0        (9)


from here (9) can have its terms collected and rearranged to give the equation in terms of dy/dx as desired.

dy/dx(48y2 + 9x) = 54-18xy

dy/dx=(54-18xy)/(48y2 + 9x)           (10)


(note this can be further simplified by noticing that all constants in the expression are divisible by 3)

final answer would be:

dy/dx=(18-6xy)/(16y2 + 3x)              (11)

Jake S. GCSE Maths tutor, A Level Maths tutor, GCSE Electronics tutor...

2 years ago

Answered by Jake, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Benjamin W.

Degree: Medicine (Bachelors) - Oxford, Balliol College University

Subjects offered: Maths, Science+ 9 more

Human Biology
History of Art
.BMAT (BioMedical Admissions)
-Personal Statements-
-Medical School Preparation-

“I am a friendly first year medical student studying at Oxford University, and having a fantastic time! I have always been interested in science, in particular human biology (the heart and the brain are my favourites) to A-level standa...”

MyTutor guarantee

£20 /hr

Jake B.

Degree: Mathematics (Masters) - Durham University

Subjects offered: Maths, Chemistry


“About Me: Hi, my name is Jake, and I am a second year Mathematics Student at Durham University. I have tutored maths for 3 years now, at both GCSE and A level, and I absolutely love it! I My Sessions: During the sessions I will cove...”

£26 /hr

Priya L.

Degree: Economics (Bachelors) - Warwick University

Subjects offered: Maths, Further Mathematics + 1 more

Further Mathematics

“About Me: I recently graduated from the University of Warwick with an Economics degree. I am currently on a gap year before I begin my graduate role as a Management Consultant in October. I decided to tutor because I wanted to spend m...”

About the author

£20 /hr

Jake S.

Degree: Mechatronics and Roboitcs (Masters) - Leeds University

Subjects offered: Maths, Physics+ 2 more

Design & Technology

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Find the derivative of x^x

Solve the equation x=4-|2x+1|

Discriminants and determining the number of real roots of a quadratic equation

How do I solve equations like 3sin^2(x) - 2cos(x) = 2

View A Level Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss