» Chapter3:
Derivatives
Section1 Derivatives
Section2 The Derivative as a Function
Section3 Differentiation formulas
Section4 Rates of Change in the natural and social Sciences
Section5 Derivatives of Trigonometric functions
Letters From Previous Calculus Students


section1
How one quantity changes in relation to another

Slope of the tangent to a curve at a point (a, f(a))


Example:
f (0) = 2
f’(-1) = -2
f’(-2)= 0
f’(-3) = 1



The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a

Page 134

#1 On the given graph of f, mark lengths that represent f (2). f( 2+h), f (2+h) – f (2), and h.
What line has slope

f (2+h) – f (2) has slope




#5 Sketch the graph of the function f for which f (0) = 0, f’(0) = 3, f’(1) = 0, f’(2) = -1.
f(0) = 0
f ‘(0) = 3
f ‘(1) = 0
f‘(2) = -1



#9 If F(x)= , find f’(1) and use it to find an equation of the tangent line to the curve at the point (1,3).

f(x) =
f’(x) = -2
(1,3)



#13 Find f’(a)

f(x) = 1 + x – 2x2
f ‘(a) = ?


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section2
Finding Derivatives Given a Function

Graph these ordered pairs to give you the graph of f’(x)
Graph these ordered pairs to give you the graph of f’(x)

Derivative of equation without the equation from page 146


Theses are all the different ways to say differentiate or derivative
f’(x) Df(x)
y’ Dxf(x)
f’(a)

A function is differentiable if you can take the derivative at A if f’(a) exists on the open interval (a,b) (a,) (-, b) (-,) if it is differentiable at every number in the interval.

Not Differentiable if:
  • Sharp turn (pointy)
  • Vertical tangent
  • Discontinuous at the point


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section3
Differential Formulas

Derivative of a constant

The Power Rule: if n is a positive integer, then



The Constant Multiple Rule: If c is a constant and f is differentiable function, then



The Sum Rule: If f and g are both differentiable, then


The Difference Rule: If f and g are both differentiable, then


Example of the difference rule and sum rule combined!!!!


The Product Rule: If f and g are both differentiable, then



The Quotient Rule: If f and g are both differentiable, then



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section4
Rates of change in natural and social sciences


The particle is at rest at t = 1, 5. The particle is moving forward when t > 5, 0 < t < 1. The total distance in 8 seconds is 120 meters. The displacement in 8 seconds is 56 m.

C(x) = cost function C ‘(x) = marginal cost


Page 167

#1 A particle moves according to a law of motion s=f(x), t>u where t is measured in seconds, and s is measured in feet.

c) The particle is at rest at 5 sec
d) The particle is moving positively at 5.1 to 6 sec
e) The distance traveled is 4 meter
f) Motion diagram


#17 The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by . Find the current when (a) t=.5 s and (b) t=1s. At what time is the current lowest?


.00000001 is the lowest possible.

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section5
Derivatives of trigonometric Functions

From the function written above it is understood that sin x means the sine of the angle whose radian measure is x. The Derivative of this function is cos x.
According to the definition of a derivative, We have:

Now,

As shown in 2.2:

Therefore,

=cosx

Derivatives of Trigonometric Functions
sin x = cos x csc x = –csc x cot x
cos x = –sin x sec x = sec x tan x
tan x = (sec) cot x = –(csc) x

example


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Letters