Page 266 - DMTH201_Basic Mathematics-1

Unit 9: Differential Calculus

9.5 Implicit Functions Notes

If a function is in the form y = f(x), then the function is said to be in the explicit form. Instead of
this, if the variables x and y are related by means of an equation, then the function is said to be
in the implicit form. In general an implicit function is given by f(x, y) = c where c is a constant.
2 x 2 y 2
e.g., y 2 4ax ,x 2 y 2 a , 1
x 2 b 2

To find the derivative of the Implicit Function f(x, y) = c

dy
Differentiate f(x, y) = c using the rules of differentiation. Collect all the terms containing on
dx
dy
the left hand side and the remaining terms on the right hand side. Take the common factor
dx
dy dy
on the left hand side. Divide both sides by the coefficient of to get .
dx dx

dy
Example: Find , if, y 2 4ax
dx
Solution: Differentiate both sides w.r.t. x

dy
2y 4 1
a
dx
dy 4a dy 2a
i.e.,
dx 2y dx y
dy
Example: Find , if, x 2 y 2 2xy
dx
Solution: Differentiate w.r.t. x
dy dy
2x 2y 2 x y
dx dx

Cancelling 2 on both sides, we get
dy dy
x y x y
dx dx
dy dy
y x y x
dx dx
dy
y x y x
dx
dy y x
dx y x
dy
1
dx

LOVELY PROFESSIONAL UNIVERSITY 259