Calculus and Analytical Geometry

Lecture # 9

MTH 104

Chain rule

1002 1xdxd

1 and 1let 21002 xuxy100then uy

Consider

xdxdu

ududy

2 and 100 99

dxdu

dudy

dxdy

)1(200

2)1(100

2100

2

2

99

xx

xx

xu

multiply rates

Chain rule

)( and ))(( xguxgfy

dxdu

dudy

dxdy

)cos( if Find 3xydxdy

If g is differentiable at x and f is differentiable at g(x) then the composition fog is differentiable at x. Moreover, if

Then y=f(u) and

Example

Alternatively

xgxgfxgfxgfdxd

Derivative of outside function Derivative of

inside function

Let 3xu cosuy Then

dudy usin

And

dxdu 23x

dxdu

dudy

dxdy

)3()sin( 2xu

32 sin3 xx

Rates of change multiply

Example

xdxd

2tan 2

tan xdxd x

dxd

x tan)tan2( xx 2sectan2

12xdxd 2

12 1x

dxd 11

21

22

12

xdxd

x

12

121

22

xx

xx

Derivative of outside function Derivative of

inside function

More examples

2 71) ( ) (3 5 )f x x x 297

5)()2

ttf

Solution 1)

xdxdu

xxu 103253Let

677Then ududy

uy

dxdu

dudy

dxdy

)103(67 xudxdy

)103(6)253(7 xxx

297

5)()2

ttf

97Let tu 7dtdu

2525 uu

y 310 ududy

dtdu

dudy

dtdy

7310 udxdy

3)97(70 tdtdy

Generalized derivative formulas

)( re whe)()( xgudxdu

ufufdxd

dxduuuu

dxd

dxduuuu

dxd

dxduuu

dxd

dxduuu

dxd

dxduuu

dxd

dxduuu

dxd

dxdurruru

dxd

cotcsccsc

tansecsec 2csccot

2sectan sincos

cossin 1

Some examples are:

Example

sin(2 )d

xdx

cos 2 2d

x xdx

2cos 2x

13

3

12 3

2 3d d

x xdx x x dx

2

3 32 3 2 3d

x x x xdx

23 22 3 3 2x x x

2

23

3 22 3x

x x

1.

2.

Example Find

2

2

2 if sin(3 )

d yy x

dx

Solution 2sin(3 )

dy dx

dx dx

2cos(3 )x 23d

xdx

26 cos(3 )x x

2

2

26 cos(3 )

d y dx x

dx dx

2 26 cos 3 cos(3 ) 6d d

x x x xdx dx

2 26 cos 3 cos(3 ) 6d d

x x x xdx dx

2 26 sin(3 ) 3d

x x xdx

26cos(3 )x

2 26 6 sin(3 ) 6cos 3x x x x

2

2 2 2

236 sin 3 6cos 3

d yx x x

dx

Example Differentiate 1ln 2 xy

1ln 2 xy

)2(1

12

xx

y

1

22

x

xy

Use the Chain Rule

Example Differentiatex

xy

ln

x

xy

ln Use the Quotient

Rule

2ln

1)1(ln

x

xxx

y

2ln

1ln

x

xy

Related rates

Consider a water is draining out of a conical filter. The volume V, the height h and the radius r are all functions of the elapsed time t.

hrV 2

3

rate of change of dV

Vdt

2[ ]3

dV dr h

dt dt

2[2 ]3

dV dr dhhr r

dt dt dt

The rate of change of V is related to the rates of change both r and h

Ralated rates

problem

Volume formula:

Example Suppose that x and y are differentiable functions of t and are related by the equation . Find dy/dt at time t=1 if x=2 and dx/dt=4 at time t=1.

3y x

solution

3y xDifferentiating both sides with respect to t

3dy dx

dt dt

23dx

xdt

2

1 1

3(2)

t t

dy dxdt dt

12 4 48

t=1 t=1

2, 4Known: Unknown : dx dydt dt

x

Example Suppose x and y are both differentiable functions of t and are related by the equation y = x2 + 3. Find dy/dt, given that dx/dt = 2 when x = 1

Solution

y = x2 + 3

Given dx/dt = 2 when x = 1

To find dy/dt

]3[][ 2 xdt

dy

dt

d

dt

dxx

dt

dy2

1 1

2(1) 2 2 4

x x

dy dxdt dt

Procedure for solving related rates problems

Step 1. Assign letters to all quantities that vary with time and any others that seem relevant to the problem. Give a definition for each letter.

Step 2. Identify the rates of change that are known and the rates of change that is to be found. Interpret each rate as a

derivative.Step 3. Find an equation that relates the variables whose rates of

change were identified in Step 2. To do this, it will often be helpful to draw an appropriately labeled figure that

illsutrates the relationship.Step 4. Differentiate both sides of the equation obtained in Step 3

with respect to time to produce a relationship between the known rates and the unknown rates of change.

Step 5. After completing Step 4, substitute all known values for the rates of change and the variables, and then solve for the unknown rate of change.

Example Suppose that , where both x and y are changing with time. At a certain instant when x=1 and y=2, x is decreasing at the rate of 2 units/s, and y is increasing at the rate of 3 units/s. How fast is z changing at this instant? Is z increasing or decreasing?

3 2z x y

solution

1 12 2

2, Known 3 :x xy y

dx dydt dt

12

Unknown: xy

dzdt

3 2z x y

Differentiating with respect to t

3 2dz dx y

dt dt

2 3 3 2dz d dy x x y

dt dt dt

2 2 33 2dx dy

x y x ydt dt

2 2 3

1 1 12 2 2

3(1) (2) 2(1) (2)x x xy y y

dz dx dydt dt dt

12

12( 2) 4(3) 12 units/sxy

dzdt

Negative sign shows that it is decreasing

Example A stone is dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3ft/s. How rapidly is the area of enclosed by the ripple increases at the end of 10 s?

Solution

let r= radius of circular ripple, A= Area enclosed by the ripple

10

dr 3, To find

dtr

GivendAdt

Since radius is increasing with a constant rate of 3 ft/s, so after 10 s the radius will be 30 ft.

We know that 2A r

2dA dr

rdt dt

2

10

2 (30)(3) 180 /

r

ft sdAdt