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Differentiation Rules (3.1Derivatives of Polynomials and Exponential…
Differentiation Rules
3.1Derivatives of Polynomials and Exponential Function
New Derivative from Old
The Constant Multiple Rule
The Sum Rule
The Difference Rule
Exponential Function
Derivative of the Natural Exponential Function
Power Function
Derivative of a Constant Function
3.6
Derivatives of Logarithmic
Functions
Logarithmic Differentiation
The method used in the next example is called logarithmic
differentiation.
The Number e as a Limit
3.3 Derivatives of Trigonometric
Functions
3.4 The Chain Rule
3.2 The Product and Quotient Rules
The Quotient Rule
The Product Rule
3.7
Rates of Change in the
Natural and Social Sciences
The difference quotient
is the average rate of change
of y with respect to x over the
interval [x1, x2]
Its limit as Δx → 0 is the derivative f ′(x1), which can
therefore be interpreted as the instantaneous rate of
change of y with respect to x or the slope of the tangent
line at P (x1, f (x1)).
Physics
If s = f (t) is the position function of a particle that is moving
in a straight line, then Δs/Δt represents the average velocity
over a time period Δt, and v = ds/dt represents the
instantaneous velocity (the rate of change of displacement
with respect to time).
The instantaneous rate of change of velocity with respect to
time is acceleration: a (t) = v ′(t) = s″ (t).
3.8Exponential Growth and
Decay
Equation 1 is sometimes called the law of natural growth
(if k > 0) or the law of natural decay (if k < 0). It is called a
differential equation because it involves an unknown
function y and its derivative dy /dt.
3.9Related Rates
If we are pumping air into a balloon, both the volume and
the radius of the balloon are increasing and their rates of
increase are related to each other.
But it is much easier to measure directly the rate of
increase of the volume than the rate of increase of the
radius.
In a related rates problem the idea is to compute the rate of
change of one quantity in terms of the rate of change of
another quantity (which may be more easily measured).
The procedure is to find an equation that relates the two
quantities and then use the Chain Rule to differentiate both
sides with respect to time.
