1. Differentiation Of Sums And Differences
Let f and g
be differentiable at x. Then f + g
and f g are differentiable at x,
The proof that ( f g)'(x)
= f '(x) g'(x) is similar.
In words, the derivative of a sum
is the sum of the derivatives, and the derivative of a difference is the
difference of the
2. Differentiation Of Constant Functions
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.
A constant function.
If f(x) = c for all x, where c is a constant, then f '(x) = 0 for all x.
For any point x we have:
3. Differentiation Of Constant Multiples Of Functions
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).
Let f be differentiable at x and c a constant. Then cf is also differentiable at x, and:
(cf )'(x) = c . f '(x).
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) (an1 + an2 b + an3b2 + ... + a2bn3 + abn2 + bn1),
which can be checked by multiplying out the right-hand side.
Let n be a positive integer. Then:
Using the factorization of the difference of nth powers we get:
It may be easier to memorize the formula in this theorem in this form: (xn)' = nxn1.
5. Differentiation Of Polynomials
Theorems 1-4 show that the derivative of the polynomial:
P(x) = a0 + a1x + a2x2 + a3x3 + ... + an1xn1 + anxn
P '(x) = a1 + 2a2x + 3a3x2 + ... + (n 1)an1xn2 + nanxn1.
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, its 1, like the
subscript of a1; remember, x1 = x. So the polynomial P(x) can be written in concise form as follows:
Find the derivative of y = x4 2x3 + 3x2 4x + 5.
y' = 4x3 6x2 + 6x 4.
Calculate f '(2) if f(x) = 3x2 5x.
We have f '(x) = 6x 5. Hence f '(2) = 6(2) 5 = 7.
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).
f(x) = 6x 12x2 15 + 30x = 12x2 + 36x 15,
f '(x) = 24x + 36.
2. Find (d/dt) g(t) if:
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.
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.
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).
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,
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.