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The limit of these average rates of change is called the (instantaneous)
rate of change of y with respect to x at x = x
1
, which is interpreted
as the slope of the tangent to the curve y = f(x) at P (x
1
,f(x
1
)):
inst.rate of change = lim
∆x→0
∆y
∆x
= lim
x
2
→x
1
f(x
2
) − f(x
1
)
x
2
− x
1
.
We recognize this limit as being the derivative of f(x) at x
1
,thatis
f
(x). This gives one more interpretation of the derivative:
The derivative f
(a) is the instantaneous rate of change of y =
f(x) with respect to x when x = a.
The velocity of a particle is the rate of change of displacement
with respect to time. Physicists are interested in other rates of change
as well – for instance, the rate of change of work with respect to time
(which is called power). Chemists who study a chemical reaction are
interested in the rate of change in the concentration of a reactant
with respect to time (called the rate of reaction). A steel manufacturer
is interested in the rate of change of the cost of producing x tons of
steel per day with respect to x (called the marginal cost). A biologist
is interested in the rate of change of the population of a colony of
bacteria with respect to time. In fact, the computation of rates of
change is important in all of the natural sciences, in engineering, and
even in social sciences.
4.6. Derivatives of elementary functions
Derivative of a constant is zero:
d
dx
(c)=0.
The Power Rule:
d
dx
(x
n
)=nx
n−1
,heren is any real number.
The Constant Multiple Rule:
d
dx
(cy)=c
dy
dx
, y = f(x).
The sum rule: [f(x)+g(x)]
= f
(x)+g
(x).
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