- SUCCESSIVE DIFFERENTIATION.
Let us try the effect of repeating several times over the operation of differentiating a function (see p. 13). Begin with a concrete case.
Let y = x5.
First differentiation, 5×4.
2 2
Second differentiation, 5 × 4×3 = 20×3. Third differentiation, 5 × 4 × 3x = 60x . Fourth differentiation, 5 × 4 × 3 × 2x = 120x.
Fifth differentiation, 5 × 4 × 3 × 2 × 1 = 120. Sixth differentiation, = 0.
There is a certain notation, with which we are already acquainted (see p. 14), used by some writers, that is very convenient. This is to employ the general symbol f (x) for any function of x. Here the symbol f ( ) is read as “function of,” without saying what particular function is meant. So the statement y = f (x) merely tells us that y is a function of x, it may be x2 or axn, or cos x or any other complicated function of x.
The corresponding symbol for the differential coefficient is f′(x),
dy
which is simpler to write than
of x.
. This is called the “derived function”
dx
Suppose we differentiate over again, we shall get the “second derived function” or second differential coefficient, which is denoted by f ′′(x); and so on.
Now let us generalize. Let y = f (x) = xn.
First differentiation, f′(x) = nxn−1.
Second differentiation, f ′′(x) = n(n − 1)xn−2.
Third differentiation, f ′′′(x) = n(n − 1)(n − 2)xn−3. Fourth differentiation, f ′′′′(x) = n(n − 1)(n − 2)(n − 3)xn−4.
etc., etc.
But this is not the only way of indicating successive differentiations.
For,
if the original function be y = f (x);
dy
dx
once differentiating gives
d dy
= f′(x);
twice differentiating gives
dx = f′′(x); dx
d2y
d2y
and this is more conveniently written as , or more usually .
(dx)2 dx2
Similarly, we may write as the result of thrice differentiating,
f ′′′(x).
d3y dx3 =
CALCULUS MADE EASY 50
Examples.2
Now let us try y = f (x) = 7×4 + 3.5×3 − 1 x2 + x − 2.
dy
—
= f′(x) = 28×3 + 10.5×2 x + 1,
dx
d2y
dx2 = f
′′(x) = 84×2
+ 21x − 1,
d3y
dx3 = f
′′′
(x) = 168x + 21,
d4y
dx4 = f
′′′′
(x) = 168,
d5y
dx5 = f
′′′′′
(x) = 0.
In a similar manner if y = ϕ(x) = 3x(x2 − 4),
ϕ′(x) = dy
dx
= 3 x × 2x + (x2 − 4) × 1 = 3(3×2 − 4),
ϕ′′
d2y
(x) = dx2 = 3 × 6x = 18x,
ϕ′′′
d3y
(x) = dx3 = 18,
ϕ′′′′
d4y
(x) = dx4 = 0.
Exercises IV. (See page 253 for Answers.)
Find
dy
and
dx
d2y
dx2 for the following expressions:
y = 17x + 12×2.
y =
x2 + a
.
x + a
y = 1 +
x x2
+ +
1 1 × 2x3
+
1 × 2 × 3
x4
.
1 × 2 × 3 × 4
Find the 2nd and 3rd derived functions in the Exercises III. (p. 45), No. 1 to No. 7, and in the Examples given (p. 40), No. 1 to No. 7.
