Maidstone Grammar School for Girls ~ A level Mathematics (C4)
L.O.: To find derivatives for functions defined implicitly.
To use implicit differentiation to be able to find the gradient of a curve, the equation of a tangent
and to locate turning points.
Functions can be defined explicitly or implicitly:-
Explicit functions have the form y = f(x) Implicit functions do not have y as the
222 xy？，9 yx，？9
2325x？ xy，36 y，xe
33 yx，？5sin(2)！ xyxy？，
22Not all functions can be expressed in the form y = f(x), whilst other functions (such as ) xy？，9are simpler written implicitly.
22Find the gradient of the curve at the point . xy？，92,5;；
22We can differentiate term by term:- xy？，9
Easy to differentiate with How do we differentiate Easy to differentiate 2 with respect to x? with respect to x. respect to x. y
2We can use the chain rule to differentiate with respect to x:- y
22Therefore, when you differentiate , you get: xy？，9
dySo at the point , 2,5，;；dx
d3 = (23)xy？dx
dy23Example: Find an expression for on the curve xxy？？，22dx
Differentiating term by term:
dySo = dx
32，Example 2: Find for the implicit function yxxyy？，25
Applications to tangents, normals and turning points
22Example 1: A curve has equation . xxyy？？，311
a) Find the equation of the tangent to the curve at the point (1, 2).
b) Show that the normal to the curve at (1, 2) goes through the point (9, 9).
a) Differentiating the terms in the equation with respect to x gives:
Therefore at the point (1, 2) the gradient is:
The equation of a tangent is . yymxx？，？()11
The gradient m =
So the equation is:
b) The gradient of the normal at (1, 2) is:
The equation of the normal is yymxx？，？()11
So the equation of the normal is
When x = 9, we find that y =
This is as required.
22Example 2: Find the coordinates of all the stationary points on the curve xyxy？？，3
Plenary: Examination style question
22a) Find the gradient to the curve at the point (2, 0). xxyyxy？？，？？332
b) Find the equation of the tangent to the curve at this point.
Correct the “solution” to this question given below:
a) Differentiating the terms in the equation with respect to x:
So at the point (2, 0):
b) The equation of a tangent is
y = mx + c.
The gradient, m, is ？1.