This document discusses implicit differentiation, which is a technique for finding the gradient function of implicit equations where x and y are not explicitly defined. It provides examples of implicit equations and their derivatives using implicit differentiation. The key steps are to take the derivative of every term with respect to x and apply the chain rule to terms containing y. Practice questions are provided to find the equations of the tangent and normal to implicit curves at given points.

1. MHL Implicit Differentiation 1
Implicit Differentiation
How do you find the gradient functions for the following:
These are called implicit functions because neither x or y are given explicitly.
Give some examples of explicit functions
We cannot always express an implicit function explicitly so we need a new
technique to find the gradient function.
Differentiate with respect to x
Step 1
Differentiate every term with respect to x
How do you differentiate?
Step 2
Use the chain rule for any y terms
2 2
( ) = ( )×
d d dy
y y
dx dy dx
2 3
+ = 5 ,y x
sin =1,x y
2 2
1 1
+ = xy
x y
2 3
+ = 5y x
2 3
( )+ ( ) = (5)
d d d
y x
dx dx dx
2 3
( )+ ( ) = (5)
d d d
y x
dx dx dx
= 2
dy
y
dx

2. MHL Implicit Differentiation 2
This gives
Now change the subject:
Practice questions
Differentiate with respect to x:
Find the equation of the tangent to the curve x2 + y2 – xy = 7 at the point (2, –1).
Find the equation of the normal to the curve 2x2 – (y + 3)2 = 1 at the point (5, 4).
3 2
2 +7 5 = 8x y y x 
dy
dx
2
+ 32 0=
dy
xy
dx