Winter wk 6 – Tues.8.Feb.05

• Calculus Ch.3 review: – Polynomial rule for derivatives

– Differentiating exponential functions

– Chain rule and product rule

• 3.5 Trigonometric functions

• 3. 6 Applications of chain rule

Energy Systems, EJZ

Differentiating polynomials and ex

1( )nn ndf d xIf f x then n x

dx dx

Differentiating polynomials:

Integrating polynomials:

Slope of ex increases exponentially:

d/dx(ex) = ex

d/dx(ax) = ln(a) ax

1

1

pp p x

If g x then g dx x dxp

Review chain rule

Differentiate exponential function y=eax

Differentiate y=ex2=ez z=x2

___________ , ______________

ax zy e e where z ax

dy dz

dz dxdy dy dz

dx dz dx

___________ , ______________dy dz

dz dxdy dy dz

dx dz dx

Review product rule and quotient rule

If y(x) = f(x) g(x) then

Practice: y = x2 e-3x=f.g where f=______, g=______

If y(x) = f(x) / g(x) then

dy dg dff g

dx dx dx

2

' ''f g g f

yg

2 -3xdf dx dg de= ____________, = _____________

dx dx dx dxdy

dx

Differentiating trig functions

Sketch the slope of y=sin(x)

Does this look familiar? sin( ) ___________d

xdx

Differentiating cosine

Sketch the slope of y=cos(x)

Does this look familiar? cos( ) ___________d

xdx

Conceptest 1

Conceptest 1 solution

Conceptest 2

Conceptest 2 solution

Conceptest 3(Hint: where is df/dx=0?)

Conceptest 3 solution

Conceptest 4(Hint: calculate the slope)

Conceptest 4 solution

Differentiating tangent

tan(x) = sin(x)/cos(x)1. Use identity sin2x+cos2x=1 to derive tan2x+1=_____

2. Use product or quotient rule to find d(tan(x))/dx

Practice differentiating trig functions

Drill on Ch.3.5 odd # problems through 41 (p.131)

(skip #19)

#5: y = sin(3x) = sin z where z=__________(sin ) (3 )

___________ , ______________

(sin(3 ))_______________________

dy d z dz d x

dz dz dx dx

d x dy dy dz

dx dx dz dx

3.6: Applications of the chain rule

Finding the derivative of an inverse function (133)

What is df/dx if f=x½? Trick: Write f2=x

Differentiate

Solve for df/dx=

(Compare to result from polynomial rule.)

2( )

d df x x

dx dx

Using chain rule to find d/dx(lnx)

Recall that elnx=x. Differentiate

Practice p.136 on odd problems through #15

ln xd dx e

dx dx

ln

ln

( )ln

( )______________ ________________

____ ______________________

ln

vx v

v

x

d d d e dve e where v x

dx dx dv dx

d e dv

dv dxdx d

edx dx

dx

dx

Using chain rule to find d/dx(ax)

Recall that ln(ax)=x ln a. Differentiate

Practice p.136 odd problems thru #35 (skip arc_ probs)

(ln )xd

adx

(ln )ln ln

(ln )______________ ________________

ln ___________ ln ______________

x x

x

x

d d d v dva v where v a

dx dx dv dxd v dv

dv dxd d

a x adx dx

dSolve for a

dx

Chain rule for related rates

Practice p.137 #46, 48