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1. Differentiation Of Sums And
Differences 
Let f and g
be differentiable at x. Then f + g
and f g are differentiable at x,
and: 
Proof
The proof that ( f g)'(x)
= f '(x) g'(x) is similar.
EOP
In words, the derivative of a sum
is the sum of the derivatives, and the derivative of a difference is the
difference of the
derivatives.
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2. Differentiation Of Constant Functions 
Recall
that a constant function is one that is, well, constant, ie, one whose values
at all points in its domain are the same.
The graph of such a function is a horizontal line, as shown in Fig. 2.1. Since
a constant function doesn't change, its rate
of change, or derivative, is 0.

Fig. 2.1 A constant function. 
Theorem 2.1
If f(x) = c for all x, where c is a constant, then f '(x) = 0 for all x. 
Proof
For any point x we have:
EOP
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3. Differentiation Of Constant Multiples
Of Functions 
Let
f(x) be a function and c a constant. The function cf defined by (cf
)(x) = c . f(x) is a constant multiple of f(x). Note
that a constant multiple of f(x) isn't a constant function unless f(x) is. We'll see in the
following theorem that the
derivative of c . f(x) equals c multiplied by the derivative of f(x).
Theorem 3.1
Let f be differentiable at x and c a constant. Then cf is also differentiable at x, and: (cf )'(x) = c . f '(x). 
Proof
In words, the derivative of a constant multiple is the constant multiple of the derivative.
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4. Differentiation Of Positive Integer
Powers Of x 
The proof of the following theorem uses the factorization of the difference of nth powers:
a^{n} b^{n} = (a b) (a^{n}^{1} + a^{n}^{2}^{ }b + a^{n}^{3}b^{2} + ... + a^{2}b^{n}^{3} + ab^{n}^{2} + b^{n}^{1}),
which can be checked by multiplying out the righthand side.
Theorem 4.1
Let n be a positive integer. Then:

Using
the factorization of the difference of nth powers we get:
EOP
It may
be easier to memorize the formula in this theorem in this form: (x^{n})' = nx^{n}^{1}.
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5. Differentiation Of Polynomials 
Theorems 14 show that the derivative of the polynomial:
P(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + ... + a_{n}_{1}x^{n}^{1} + a_{n}x^{n}
is:
P '(x) = a_{1} + 2a_{2}x + 3a_{3}x^{2} + ... + (n 1)a_{n}_{1}x^{n}^{2} + na_{n}x^{n}^{1}.
Why
isn't there the factor x in the zeroth term a_{0} of P(x)? Well, there
is, but its exponent is 0, like the subscript of the
factor a_{0}, and x^{0} = 1, so that a_{0}x^{0} = (a_{0})(1) = a_{0}. What's the exponent of x in the first term a_{1}x? Well, its 1, like the
subscript of a_{1}; remember, x^{1} = x. So the polynomial P(x) can be written in concise form as follows:
Example 5.1
Find the derivative of y = x^{4} 2x^{3} + 3x^{2} 4x + 5.
Solution
y' = 4x^{3} 6x^{2} + 6x 4.
EOS
Example 5.2
Calculate f '(2) if f(x) = 3x^{2} 5x.
Solution
We have f '(x) = 6x 5. Hence f '(2) = 6(2) 5 = 7.
EOS
f '(2) is the derivative of f(x)
at x = 2, or the value of f '(x)
at x = 2, not the derivative of f(2) (the derivative of f(2) is 0
because f(2) is a constant).
That's why we calculate f '(x) first,
then we substitute x = 2 in f '(x).
Problems & Solutions 
1. Differentiate f(x) = (2x 5)(3 6x).
Solution
f(x) = 6x 12x^{2} 15 + 30x = 12x^{2} + 36x 15,
f '(x) = 24x + 36.
2. Find (d/dt) g(t) if:
Solution
3. Let f(x) = x(3x 5)^{2}
and I = [1, 3].
a. Find the average rate of change of f over I.
b. Find the instantaneous rate of change of f at the midpoint of I.
Solution
Now:
f(x) = x(3x 5)^{2} = x(9x^{2} 30x + 25) = 9x^{3} 30x^{2} + 25x.
So f '(x) = 27x^{2} 60x + 25. Thus, the instantaneous rate of change of f at the midpoint of I is:
f '(1) = 27(1)^{2} 60(1) + 25 = 8.
4. Let C
be the curve with equation y = 4x^{2} x^{4}.
a. Find all the horizontal tangent lines to C.
b. Find the normal line to C at x
= 1/2.
Solution
a. 

Note
Recall that the notation y(a) means the value of y at x = a, not the product ya.
5. The equation xy' = 3y involves the derivative of a function
and thus is called a differential equation.
a. Show that for any constant C, the function y = Cx^{3} satisfies the given equation. That
function is called a solution of
the given differential
equation.
b. Determine the particular solution y = f(x) of the given differential equation that passes thru the
point (2, 5).
Solution
a. From y = Cx^{3} we have y' = 3Cx^{2}.
So, xy' = x(3Cx^{2}) = 3(Cx^{3}) = 3y, which shows that y = Cx^{3}
satisfies the equation
xy' = 3y.
b. 5 = f(2)
= C(2)^{3} = 8C, thus C = 5/8. Consequently, y = (5/8)x^{3} is the particular solution that passes
thru the point (2,
5).
6. Suppose that when a pebble is dropped into a
tank of water, a wave travels outward in a circular ring whose radius
increases at a constant rate of 20
cm/sec.
a. Find the area of the circular disk enclosed
by the wave 3 seconds after the pebble hits the water.
b. Find the instantaneous rate at which the area of the disk is
increasing 3 seconds after the pebble hits the water.
Solution
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