Chain Rule in Partial Differentiation

PARTIAL DIFFERENTIATION SEBASTIAN VATTAMATTAM

1. Partial Derivatives and Chain Rules Definition 1.1. Let f be a function of several variables.

Its derivative with respect to one of those variables, keeping others constant, is called a partial derivative.

The partial derivative of function f with respect to x is denoted by ∂f ∂x or fx .

Definition 1.2. Let f be a function of x, y. ∂f ∂f ∂x , ∂y are called the first order partial derivatives of f.

See Figure 1 The second order partial derivatives are ∂ 2f ∂x2 ∂ 2f ∂y 2 ∂ 2f ∂x∂y ∂ 2f ∂y∂x

∂ ∂f ( ) ∂x ∂x ∂ ∂f = ( ) ∂y ∂y ∂ ∂f = ( ) ∂x ∂y ∂ ∂f = ( ) ∂y ∂x =

Example 1.3. f (x, y) = x2 sin y 1

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SEBASTIAN VATTAMATTAM

Figure 1. First Order Partial Derivatives

∂f ∂x ∂f ∂y ∂ 2f ∂x2 ∂ 2f ∂x∂y ∂ 2f ∂y∂x ∂ 2f ∂y 2

= 2x sin y = x2 cos y = 2 sin y = 2x cos y = 2x cos y = −x2 sin y

PARTIAL DIFFERENTIATION

Theorem 1.4. ∂ 2f ∂ 2f = ∂x∂y ∂y∂x Theorem 1.5. Chain Rule 1 If z = f (x, y) and x = x(s, t), y = y(s, t) then ∂z ∂x ∂z ∂y ∂z = + ∂s ∂x ∂s ∂y ∂s ∂z ∂z ∂x ∂z ∂y = + ∂t ∂x ∂t ∂y ∂t See Figure 2

Figure 2. Chain Rule 1

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SEBASTIAN VATTAMATTAM

Example 1.6.

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z = log(u2 + v); u = ex+y , v = x2 + y ∂z ∂u ∂z ∂v ∂u ∂x ∂u ∂y ∂v ∂x ∂v ∂y

2u u2 + v 1 = 2 u +v =

= ex+y

2

= 2yex+y

2

= 2x = 1

∂z ∂z ∂u ∂z ∂v 2u x+y2 1 2 2 = + = 2 e + 2 2x = 2 (uex+y +x) ∂x ∂u ∂x ∂v ∂x u + v u +v u +v ∂z ∂u ∂z ∂v 2u 1 1 ∂z 2 2 = + = 2 2yex+y + 2 = 2 (4uyex+y +1) ∂y ∂u ∂y ∂v ∂y u +v u +v u +v

Theorem 1.7. Chain Rule 2 If z = f (x, y) and x = x(t), y = y(t) then ∂z dx ∂z dy dz = + dt ∂x dt ∂y dt See Figure 3 Example 1.8. u = sin(x/y); x = et , y = t2

PARTIAL DIFFERENTIATION

Figure 3. Chain Rule 2

∂u 1 = cos(x/y) ∂x y ∂u −x = cos(x/y) ∂y y2 dx = et dt dy = 2t dt du ∂u dx ∂u dy 1 −x = + = cos(x/y)et + 2 cos(x/y)2t dt ∂x dt ∂y dt y y t 2 Substituting x = e , y = t , and simplifying we get et du t − 2 = 3 cos( 2 ) dt t t

Theorem 1.9. Chain Rule 3

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SEBASTIAN VATTAMATTAM

If z = f (x, y) and y = g(x) then dz ∂z ∂z dy = + dx ∂x ∂y dx See Figure 4

Figure 4. Chain Rule 3

Example 1.10. z = x2 +

√

y; y = sin x

∂z = 2x ∂x 1 ∂z = √ ∂y 2 y dy = cos x dx dz ∂z ∂z dy 1 cos x = + = 2x + √ cos x = 2x + √ dx ∂x ∂y dx 2 y 2 sin x Mary Bhavan, Ettumanoor E-mail address: vattamattam@gmail.com