Transcript Differentiation by First Principles

Differentiation by
First Principles
© Annie Patton
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Aim of Lesson
To learn, what is meant by differentiation by
First Principles and to see how to do this
in five steps.
© Annie Patton
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5 steps to Differentiate by first principles
Step 1: Given f(x)
Step 2: Find f(x+h)
Step 3: Find f(x+h)-f(x)
Step 4: Find f(x+h)-f(x)
h
Step 5: Find
f(x+h)-f(x) df ( x)
lim

 f '( x)
h 0
h
dx
© Annie Patton
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Find from first principles the derivative of x2 with respect to x.
Leaving Certificate 2005 Higher Level Paper 1 no 7(a)
Step 1:
f(x)=x2
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Step 2: f(x+h)=(x+h)2= x2 +2xh+h2
Step 3: f(x+h) - f(x)= x2 +2xh+h2-x2
Step 4:
f(x+h)-f(x)
=2x+h
h
2
f(x+h)-f(x) d ( x )

 2x
Step 5 lim
h 0
h
dx
© Annie Patton
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Differentiate y=x2 +2x+4 with respect to x.
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answer.
2
f(x)=x +2x+4
2
2
2
f(x+h)=(x+h) +2(x+h)+4=x +2xh+h +2x+2h+4
f(x+h)-f(x)=x +2xh+h +2x+2h+4-x  2 x  4
2
2
f(x+h)-f(x)=2xh+h 2 +2h
2
f(x+h)-f(x)
=2x+h+2
h
dy
f(x+h)-f(x)
 lim
=2x+2
dx h 0
h
© Annie Patton
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Differentiate
1
x
with respect to x from first
principles.
Leaving Certificate 2007 Higher Level Paper 1 no 6(b)(i)
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1
f(x)=
x
1
f(x+h)=
x+h
1 1 x-(x+h)
-h
f(x+h)-f(x)=
- =
=
x+h x x(x+h) x(x+h)
f(x+h)-f(x)
-1
=
h
x(x+h)
1
d( )
x  lim f(x+h)-f(x)  -1
2
h

0
dx
h
x
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1
Find the equation of the tangent to y= at the point
x
 1
 2,  .
 2
Leaving Certificate 2007 Higher Level Paper 1 no 6(b)(i)
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1
d( )
1
x
From the last slide we know
 2
dx
x
1
1
1
So the slope of the tangent at (2, )equals- 2 =- .
2
(2)
4
1 1
Equation of the tangent is y- =- (x-2)
2 4
4y-2=-x+2
x+4y-4=0
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Findfromfirst principles thederivativeof x with respect to x.
Leaving Certificate 2001 Higher Level Paper 1 no 6(b)(i)
Start clicking when you want to see the answer.
f(x+h)= x+h
f(x)= x
f(x+h)-f(x)= x+h- x
f(x+h)-f(x)=
( x+h - x )( x+h + x )
=
( x+h) + x )
f(x+h)-f(x)
1
=
h
x+h + x
x+h-x
=
x+h + x
h
x+h + x
d x
f(x+h)-f(x)
1
1
 lim
=
=
dx h0
h
x+ x 2 x
© Annie Patton
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Find the slope and equation of the tangent of the curve
y=x 2 +2 at the point (1,3).
f(x+h)=(x+h)2 +2=x 2 +2xh+h 2  2
2
2
2
f(x+h)-f(x)=x +2xh+h +2-x -2=2xh+h
2
f(x+h)-f(x)
lim
=2x
h 0
h
dy
=2x
dx
dy
dy
=2x at (1,3) =2=the slope of the tangent at (1,3)
dx
dx
Equation of the tangent y-y1 =m(x-x1 )
y - 3=2(x - 1)
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Differentiate the following with respect to x fromfirst principles:
1. y=x 2 +2x
2
2. y=3x
3. y=2x+5
3
4. y=x
5. Find the slope and equation of the tangent of
2
the curve y=x +3 at the point (1,4)
© Annie Patton
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Differentiate, from first principles, cos x with respect to x.
Leaving Certificate 2004 Higher Level Paper 1 no 6(b)(ii)
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f(x)=cosx
f(x+h)=cos (x+h)
f(x+h)-f(x)=cos(x+h)-cosx
(cosA-cosB=-2sin
A+B
A-B
sin
)
2
2
x+h+x
x+h-x
h
f(x+h)-f(x)=-2sin(
)sin(
)
sin
f(x+h)-f(x)
2x+h
2
2
2
=-sin
h
f(x+h)-f(x)
sinA
lim
= - sinx. Since lim
=1.
h 0
A 0 A
h
© Annie Patton
2
h
2
d(cos x)
= - sinx
dx
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Find from first principles the derivative of sinx with
respect to x.
Leaving Certificate 1999 Higher Level Paper 1 no 6(b)
Start clicking when you want to see the answer.
f(x)=sinx
f(x+h)=sin(x+h)
f(x+h)-f(x)=sin(x+h)-sinx
x+h+x x+h-x
f(x+h)-f(x)=2cos
sin
2
2
1
1
f(x+h)-f(x)=2cos(x+ h)sin h
2
2
1
h
f(x+h)-f(x)
1
1
2
=2cos(x+ h).
.
1
h
2
h 2
2
sin
f(x+h)-f(x)
1
lim
=2cos(x+0).1. =cosx
h 0
h
2
© Annie Patton
d(sinx)
=cosx
dx
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5 steps to Differentiate by first
principles
Step 1: Given f(x)
Step 2: Find f(x+h)
Step 3: Find f(x+h)-f(x)
Step 4: Find
Step 5: Find
f(x+h)-f(x)
h
f(x+h)-f(x) df(x)
lim
=
=f '(x)
h 0
h
dx
© Annie Patton
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Proof of Product Rule by First
Principles
Leaving Certificate 2000 Higher
Level Paper 1 no 6(b)(i)
f ( x )  u ( x )v ( x )
f ( x  h)  u( x  h)v( x  h)
f ( x  h)  f ( x)  u( x  h)v( x  h)  u( x)v( x)
f(x+h)-f(x)=u(x+h)v(x+h)-u(x+h)v(x)
+u(x+h)v(x)-u(x)v(x)
f(x+h)-f(x)
(v(x+h)-v(x))
(u(x+h)-u(x))
=u(x+h)
+v(x)
h
h
h
f(x+h)-f(x)
dv
du df
lim
=u(x)
+v(x)
=
h 0
h
dx
dx dx
© Annie Patton
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Homework
Differentiate the following from first principles.
1 y  3x
2. y  x 2  5
3. y  cos x  sin x
4
4. y  2
x
5 y  3x 2  6 x
6 y  2 x 4  x  12
© Annie Patton