Techniques of differentiation
df
. This notation allows us
[Another way of writing f 0(x) is as dx
to write things like
d 2
[x − 3x + 1] = 2x − 3.
dx
The power rule:
d n
[x ] = n.xn−1.
dx
(For any real number n.)
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NOTE: This agrees with the calculations we made last time:
d 3
[x ] = 3x2,
dx
and
d 4
[x ] = 4x3.
dx
We can also use this to see:
d 1
1 −1
d √
[ x] =
[x 2 ] =
x 2
dx
dx
2
and
√ √
d √2
2−1
[x ] =
2 x
dx
and so on...
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We have some other rules:
• If c is a constant and f (x) is a differentiable function then:
d
d
[cf (x)] = c [f (c)]
dx
dx
• (The Sum Rule) If f (x) and g(x) are differentiable then
d
d
d
[f (x) + g(x)] =
[f (x)] +
[g(x)] .
dx
dx
dx
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• (The Product Rule) If f (x) and g(x) are differentiable then
d
d
d
[f (x).g(x)] =
[f (x)] .g(x) + f (x) [g(x)] .
dx
dx
dx
= f 0(x).g(x) + f (x).g 0(x).
• (The quotient rule) If f (x) and g(x) are differentiable then
whenever g(x) 6= 0 we have
g(x)f 0(x) − f (x)g 0(x)
d f (x)
=
dx g(x)
g(x)2
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• (The Chain Rule) If f (y) and g(x) are differentiable then
d
[f (g(x))] = g 0(x).f 0 (g(x)) .
dx
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Examples on board ...
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Relative Rate of Change (and Percentage Rate of Change)
The Relative Rate of Change of a function f (x) with respect
to x is:
f 0(x)
f (x)
The Percentage Rate of Change of f (x) with respect to x is:
100f 0(x)
.
f (x)
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Examples:
The population (in thousands) P of a city at time t after 1850
is given by
P (t) = 100 + 70t − 1.5t2 + 0.01t3.
What is the rate of change of the population in the year 2010?
What is the percentage rate of change of population with respect
to time in 2010?
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