Page 299 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Then, using the chain rule,
dy
dy dt dx
= provided 0
dx dx dt
dt

dy 2t 1
=
dx 3t 2
1 dy 1 1
From this we can see that when t , 0 and so t is a stationary value. When t ,
2 dx 2 2

1 1
x and y and these are the coordinates of the stationary point.
8 4
dy
We also note that when t 0, is infinite and so the y axis is tangent to the curve at the point (0, 0).
dx

11.3 Second Derivatives

2
d y
Example: Suppose we wish to find the second derivative when
dx 2
x = t 2 y = t 3

Differentiating we find
dx dy
= 2t = 3t 2
dt dt
Then, using the chain rule,

dy
dy dx
= dt provided 0
dx dx dt
dt

So that

dy 3t 2 3t
=
dx 2t 2

2
d y
We can apply the chain rule a second time in order to find the second derivative, .
dx 2
2
d y d dy
=
dx 2 dx dx

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