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Derivative...

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Notation for Differentiation There are two main types of notation used to denote the derivative of a function. Lagrange’s Notation is to write the derivative of the function f (x) as f ′ (x) Leibniz’s Notation is to write the derivative of the function f as df dx

Two other notations are worth mentioning Newton’s Notation is to write the derivative of y using a dot y˙

Euler’s Notation is to use a capital D i.e. Dx f (x)

The Lagrange and Leibniz notation will be considered in some situations involving differentiation. It may be that the comments are influenced too much by the particular methods of teaching received by the author. Any further comments are welcome.

1. Functions of a single variable (a) Basic Lagrange

Leibniz

Function

f (x)

f

Derivative

f ′ (x)

df dx

2nd Derivative

f ′′(x)

d2 f dx2

Higher Derivative

Integral

f

(n)

dn f dxn

(x)

Z

f (x) dx

Comments For the higher derivatives the (n) is a little cumbersome and can possible be mistaken for an index. An integral is rarely seen without a dx so there is no entry in the Lagrange Column.

(b) Differentiation Rules

Product Rule Lagrange

[u(x)v (x)]′ = u(x)v ′ (x) + v (x)u′ (x)

Leibniz

d dv du [uv] = u + v dx dx dx

Comments The Leibniz notation is probably more common here.

Chain Rule Lagrange

f [g (x)]′ = f ′ [g(x)] × g ′ (x)

Leibniz

df dg d [f (g(x))] = × dx dg dx

Comments Neither set looks ’comfortable in its entirety. The most d comfortable may be a mixture such as [f (g(x))] = f ′ [g(x)] × g ′ (x) dx Implicit Differentiation (say of x2 f + sin(xf ) = 0) Lagrange

x2 f ′ (x) + 2x f (x) + cos[x f (x)][f (x) + xf ′ (x)] = 0

Leibniz

x2

df df + 2xf + cos(xf )(f + x ) = 0 dx dx

Comments The Leibniz notation looks more natural here.

Logarithmic Differentiation (say of f = (x + 1)5 (sin x)7 (x2 − 1)4 ) Lagrange

f ′ (x) 5 8x = + 7 cot x + 2 x −1 f (x) x+1

Leibniz

5 8x 1 df = + 7 cot x + 2 f dx x + 1 x −1

Comments Not a great deal to choose between them here.

Parametric Differentiation (say of f = t + sin t, t2 − cos t) Lagrange

f ′ (t) = 1 + cos t; x′ (t) = 2t + sin t f ′ (x) =

Leibniz

1 + cos t 2t + sin t

dx df = 1 + cos t; = 2t + sin t dt dt df 1 + cos t = dx 2t + sin t

Comments The Leibniz notation seems to cope better with the different variables.

(c) Integration Integration by parts Lagrange

Z

u(x)v (x) dx = u(x)v (x) −

Leibniz

Z

dv u dx = u × v − dx

′

Z

v

Z

v (x)u′ (x) dx

du dx dx

Comments The Lagrange notation is certainly more common here.

Arc Length Lagrange

Z bp 1 + [f ′ (x)]2 dx a

Leibniz

Z

a

b

s

1+

µ

df dx

¶2

dx

Comments Not a great deal to choose between the two.

(d) Other Differential Equations Lagrange

f ′ (x) + 2xf (x) = x2 sin x

Leibniz

df + 2xf = x2 sin x dx

Comments The Leibniz notation is more common here. The application to separable equations may be seen as a gimic by some.

Taylor Series Lagrange

Leibniz

f (x) = f (a) + (x − a)f ′ (a) + 21 f ′′(a) + . . . ¯ ¯ 2 ¯ d f df ¯¯ + 1 (x − a)2 2 ¯¯ f = f |x=a + (x − a) ¯ +... dx x=a 2 dx x=a

Comments The Lagrange notation is certainly more common here.

2. Functions of Two Variables (a) Basic

Function

Derivative

2nd Derivative

Higher Derivative

Lagrange

Leibniz

f (x, y)

f

fx , fy

∂f ∂f , ∂x ∂y

fxx , fxy , fyy

∂ 2f ∂ 2f ∂ 2f , , ∂x2 ∂xy ∂y 2

fxxxyy

∂ 5f ∂x3 y 2

Comments A long series of subscripts can start to look a bit clumsy.

(b) Other Chain Rule Lagrange

f ′ (x) = f ′ (u)u′ (x) + f ′ (v )v ′ (x)

Leibniz

∂f ∂u ∂f ∂v ∂f = + ∂x ∂u ∂x ∂v ∂x

Comments Both notations are used commonly

Jacobian (in double integral) Lagrange

x′ (u)y ′ (v) − x′ (v )y ′ (u)

Leibniz

∂x ∂y ∂x ∂y − ∂v ∂v ∂v ∂u

Comments The Liebniz notation is certainly more common here.

Taylor Series Lagrange

f (x, y) = f (x0 , y0 ) + (x − x0 )fx (x0 , y0 ) + (y − y0 )fy (x0 , y0 )+ 1 2

(x − x0 )2 fx x(x0 , y0¤) + (x − x0 )(y − y0 )fx y(x0 , y0 )+ +(y − y0 )2 fy y(x0 , y0 ) + . . . Leibniz

£

¯ ¯ ∂f ¯¯ ∂f ¯¯ + + (y − y0 ) f = f |x=x0 ,y=y0 + (x − x0 ) ∂y ¯x=x0 ,y=y0 ∂x ¯x=x0 ,y=y0 "

¯ ¯ ∂ 2 f ¯¯ ∂ 2 f ¯¯ (x − x0 ) + 2(x − x0 )(y − y0 ) ∂x2 ¯x=x0 ,y=y0 ∂xy ¯x=x0 ,y=y0 # ¯ 2 ¯ ∂ f +(y − y0 )2 2 ¯¯ +... ∂y x=x0 ,y=y0

1 2

2

Comments The Lagrange notation looks more comfortable here.

Partial Differential Equation Lagrange

x2 fx − 2xyfy = 1

Leibniz

x2

∂f ∂f =1 − 2xy ∂f ∂x

Comments Again, the Liebniz notation is certainly more common here.

Checking PDE Solution Lagrange

fx = 2xyg ′ (x2 y) + fy = x2 g ′ (x2 y)

Leibniz

1 x2

1 ∂f ∂g(x2 y ) + = 2xy ∂(x2 y) x2 ∂x 2 ∂f ∂g(x y ) = x2 ∂(x2 y) ∂y

Comments The Leibniz notation is having difficulty here with terms ∂g(x2 y ) such as being extremely clumsy. ∂(x2 y)

In general, the Leibniz notation rests more comfortably with these examples. However, there were several cases where the Lagrange notation had a slight advantage. For the final case of checking the solution of a partial differential equation, this was a large and significant advantage....