##### Calculus Of One Real Variable  By Pheng Kim Ving Chapter 3: Rules Of Differentiation  Section 3.1: Differentiation Of Sums, Differences, And Polynomials

3.1
Differentiation Of Sums, Differences, And
Polynomials

 1. Differentiation Of Sums And Differences

### Theorem 1.1

 Let f and g be differentiable at x. Then f + g and f  g are differentiable at x, and: ( f + g)'(x) = f '(x) + g'(x), ( f  g)'(x) = f '(x)  g'(x).

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.

 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

 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

#### EOP

In words, the derivative of a constant multiple is the constant multiple of the derivative.

 4. Differentiation Of Positive Integer Powers Of x

The proof of the following theorem uses the factorization of the difference of nth powers:

an  bn = (a  b) (an1 + an2 b + an3b2 + ... + a2bn3 + abn2 + bn1),

which can be checked by multiplying out the right-hand side.

Theorem 4.1

 Let n be a positive integer. Then:

#### Proof

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: (xn)' = nxn1.

 5. Differentiation Of Polynomials

Theorems 1-4 show that the derivative of the polynomial:

P(x) = a0 + a1x + a2x2 + a3x3 + ... + an1xn1 + anxn

is:

P '(x) = a1 + 2a2x + 3a3x2 + ... + (n  1)an1xn2 + nanxn1.

Why isn't there the factor x in the  zeroth  term a0 of P(x)? Well, there is, but its exponent is 0, like the subscript of the
factor
a0, and x0 = 1, so that a0x0 = (a0)(1) = a0. What's the exponent of x in the  first  term a1x? Well, its 1, like the
subscript of
a1; remember, x1 = x. So the polynomial P(x) can be written in concise form as follows:

Example 5.1

Find the derivative of y = x4  2x3 + 3x2  4x + 5.

Solution
y' = 4x3  6x2 + 6x  4.
EOS

Example 5.2

Calculate f '(2) if f(x) = 3x2  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  12x2  15 + 30x =  12x2 + 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(9x2  30x + 25) = 9x3  30x2 + 25x.

So f '(x) = 27x2  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 = 4x2  x4.
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 = Cx3 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 = Cx3 we have y' = 3Cx2. So, xy' = x(3Cx2) = 3(Cx3) = 3y, which shows that y = Cx3 satisfies the equation
xy' = 3y.

b.  5 = f(2) = C(2)3 = 8C, thus C = 5/8. Consequently, y = (5/8)x3 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