Chain rule for functions of 2, 3 variables (Sect. 14.4 ...

Chain rule for functions of 2, 3 variables (Sect. 14.4)

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Review: Chain rule for f : D ? R �� R.

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Functions of two variables, f : D ? R2 �� R.

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Chain rule for functions defined on a curve in a plane.

Chain rule for change of coordinates in a plane.

Functions of three variables, f : D ? R3 �� R.

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Chain rule for change of coordinates in a line.

Chain rule for functions defined on a curve in space.

Chain rule for functions defined on surfaces in space.

Chain rule for change of coordinates in space.

A formula for implicit differentiation.

Review: The chain rule for f : D ? R �� R

Chain rule for change of coordinates in a line.

Theorem

If the functions f : [x0 , x1 ] �� R and x : [t0 , t1 ] �� [x0 , x1 ] are

differentiable, then the function

f? : [t0 , t1 ] �� R given by the




composition f?(t) = f x(t) is differentiable and


dx

d f?

df

(t) =

x(t)

(t).

dt

dx

dt

Notation:

d f?

df dx

=

.

dt

dx dt




Alternative notations are f?0 (t) = f 0 x(t) x 0 (t) and f?0 = f 0 x 0 .

The equation above is usually written as

Review: The chain rule for f : D ? R �� R

Chain rule for change of coordinates in a line.

Example

The volume V of a gas balloon depends on the temperature F in

Fahrenheit as V (F ) = k F 2 + V0 . Let F (C ) = (9/5)C + 32 be the

temperature in Fahrenheit corresponding to C in Celsius. Find the

rate of change V? 0 (C ).

Solution: Use the chain rule to derivate V? (C ) = V (F (C )),

9

9

0

0

0

0

C + 32 .

V? (C ) = V (F ) F = 2k F F = 2k

5

5



9

18k

C + 32 .

We conclude that V 0 (C ) =

C

5

5

9

2

Remark: One could first compute V? (C ) = k

C + 32 + V0

9 5  9

and then find the derivative V? 0 (C ) = 2k

C + 32 .

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Chain rule for functions of 2, 3 variables (Sect. 14.4)

I

Review: Chain rule for f : D ? R �� R.

I

I

Functions of two variables, f : D ? R2 �� R.

I

I

I

Chain rule for functions defined on a curve in a plane.

Chain rule for change of coordinates in a plane.

Functions of three variables, f : D ? R3 �� R.

I

I

I

I

Chain rule for change of coordinates in a line.

Chain rule for functions defined on a curve in space.

Chain rule for functions defined on surfaces in space.

Chain rule for change of coordinates in space.

A formula for implicit differentiation.

Functions of two variables, f : D ? R2 �� R

The chain rule for functions defined on a curve in a plane.

Theorem

If the functions f : D ? R2 �� R and r : R �� D ? R2 are

differentiable, with r(t) = hx(t), y (t)i, then the function




f? : R �� R given by the composition f?(t) = f r(t) is

differentiable and holds


dx


dy

?f

?f

d f?

r(t)

r(t)

(t) =

(t) +

(t).

dt

?x

dt

?y

dt

Notation:

The equation above is usually written as

d f?

?f dx

?f dy

=

+

.

dt

?x dt

?y dt

An alternative notation is f?0 = fx x 0 + fy y 0 .

Functions of two variables, f : D ? R2 �� R.

The chain rule for functions defined on a curve in a plane.

Example

Find the rate of change of the function f (x, y ) = x 2 + 2y 3 , along

the curve r(t) = hx(t), y (t)i = hsin(t), cos(2t)i.

Solution: The rate of change of f along the curve r(t) is the

derivative of f?(t) = f (r(t)) = f (x(t), y (t)). We do not need to

compute f?(t) = f (r(t)). Instead, the chain rule implies

f?0 (t) = fx (r) x 0 + fy (r) y 0 = 2x x 0 + 6y 2 y 0 .

Since x(t) = sin(t) and y (t) = cos(2t),





f?0 (t) = 2 sin(t) cos(t) + 6 cos2 (2t) ?2 sin(2t) .

The result is f?0 (t) = 2 sin(t) cos(t) ? 12 cos2 (2t) sin(2t).

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Functions of two variables, f : D ? R2 �� R

The chain rule for change of coordinates in a plane.

Theorem

If the functions f : R2 �� R and the change of coordinate functions

x, y : R2 �� R are differentiable, with x(t, s) and y (t, s), then the

function f? : R2 �� R given
by the composition

f?(t, s) = f x(t, s), y (t, s) is differentiable and holds

f?t = fx xt + fy yt

f?s = fx xs + fy ys .

Remark: We denote by f (x, y ) the function values in the

coordinates (x, y ), while we denote by f?(t, s) are the function

values in the coordinates (t, s).

Functions of two variables, f : D ? R2 �� R

The chain rule for change of coordinates in a plane.

Example

Given the function f (x, y ) = x 2 + 3y 2 , in Cartesian coordinates

(x, y ), find the derivatives of f in polar coordinates (r , ��).

Solution: The relation between Cartesian and polar coordinates is

x(r , ��) = r cos(��),

y (r , ��) = r sin(��).

The function f in polar coordinates is f?(r , ��) = f (x(r , ��), y (r , ��)).

The chain rule says f?r = fx xr + fy yr and f?�� = fx x�� + fy y�� , hence

f?r = 2x cos(��) + 6y sin(��) ? f?r = 2r cos2 (��) + 6r sin2 (��).

f?�� = ?2xr sin(��) + 6yr cos(��),

f?�� = ?2r 2 cos(��) sin(��) + 6r 2 cos(��) sin(��).

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Chain rule for functions of 2, 3 variables (Sect. 14.4)

I

Review: Chain rule for f : D ? R �� R.

I

I

Functions of two variables, f : D ? R2 �� R.

I

I

I

Chain rule for functions defined on a curve in a plane.

Chain rule for change of coordinates in a plane.

Functions of three variables, f : D ? R3 �� R.

I

I

I

I

Chain rule for change of coordinates in a line.

Chain rule for functions defined on a curve in space.

Chain rule for functions defined on surfaces in space.

Chain rule for change of coordinates in space.

A formula for implicit differentiation.

Functions of three variables, f : D ? R3 �� R.

Chain rule for functions defined on a curve in space.

Theorem

If the functions f : D ? R3 �� R and r : R �� D ? R3 are

differentiable, with r(t) = hx(t), y (t), z(t)i, then the


function

?

?

f : R �� R given by the composition f (t) = f r(t) is

differentiable and holds

d f?

?f dx

?f dy

?f dz

=

+

+

.

dt

?x dt

?y dt

?z dt

Notation:

The equation above is usually written as

f?0 = fx x 0 + fy y 0 + fz z 0 .

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