SUMMARY OF DIFFERENTIAL NOTATION - Differentials - The Calculus Primer

The Calculus Primer (2011)

Part VII. Differentials


dc = 0

d(cv) = c dv

d(u + v) = du + dv

d(uv) = u dv + v du


d(vn) = nvn−1 dv

d(xn) = nxn−1 dx

d(loga v) = images loga e

d(log v) = images

d(av) = av log a dv

d(ev) = ev dv

d(uv) = vuv−1 du + log u·uv·dv

d(sin v) = cos v dv

d(cos v) = − sin v dv

d(tan v) = sec2 v dv

d(cot v) = − csc2 v dv

d(sec v) = sec v tan v dv

d(csc v) = − csc v cot v dv


7—6.Differentiation with Differentials. This may be illustrated by the following examples.

EXAMPLE 1.Find the differential of

y = x5 + 3x2.

Solution. dy = 5x4 dx + 6x dx,

ordy = (5x4 + 6x) dx.

EXAMPLE 2.Find the differential of


EXAMPLE 3.Find dy for y = log sin 2x.

Solution. d(log v) = images ,


EXAMPLE 4.Find dy from

a2x2 + b2y2 = a2b2.

Solution. 2a2x dx + 2b2y dy = 0,


EXAMPLE 5.Find from

ρ2 = a3 sin 3θ.

Solution. 2ρ = a3 cos 3θ·3,


7—7.Successive Differentials. Consider the function y = f(x). We may regard d(dy) as the second differential of y, or the second differential of the function; it is represented by the symbol d2y. In the same way,


is the third differential of y; and dny is the nth differential of y.

The reader must be careful in interpreting the symbolism used here. Thus

d2v is the second differential of v, and equals d(dv).

dv2 is the square of dv, and may also be written (dv)2.

d(v2) is the first differential of v2, and equals 2v dv.

In general, in the function y = f(x), the differential of the independent variable, namely, dx, is independent of x, and must be regarded as a constant when differentiating with respect to x. Consequently:

dy = f′(x)dx,

and, since dx is a constant,

d(dy) = d2y = (dx)·[f″(x) dx] = f″(x)(dx)2.

Similarly,d3y = f″′(x) (dx)3,

anddny = f(n)(x)(dx)n.

The reader should note that, since (dx)2 = dx2, the differential notation


is equivalent to the familiar derivative notation


simply by dividing both sides of (1) by (dx)2, or dx2. Similarly,



Differentiate the following, using differentials:

1. y = x3 + 5x2 − 3

2. y = (5x + 2)3

3. y = images

4. y = ketx

5. y = ex log x

6. ρ = sin2θ + 2cos θ

7. f(x) = (log x)4

8. ρ = a2 sec 3θ

Find dy for each of the following:

9. x2 + y2 = r2

10. b2x2a2y2 = a2b2

11. xy = k

Find d3y for each of the following:

12. y = x4 − 5x3 + 4x2 − 7

13. y = ex sin x

14. y = images

15. y = images