differentiation 9 examples using the product and quotient...
TRANSCRIPT
Differentiation 9examples using the product and
quotient rules
J A Rossiter
http://controleducation.group.shef.ac.uk/mathematics.html
1
Slides by Anthony Rossiter
Introduction• The previous videos have given a definition and
concise derivation of differentiation from first principles.
• The aim now is to give a number of worked examples for more challenging cases.
• Here the focus is on combining the product and quotient rules, while also utilising a table of results for simple functions.
Slides by Anthony Rossiter
2
)(
)(
xv
xuy
2v
dx
dvu
dx
duv
dx
dy
)()( xvxuy
dx
duv
dx
dvu
dx
dy
Table of common results
Slides by Anthony Rossiter
3
adx
dyaxy 1 nn nax
dx
dyaxy
)cos()sin( bxbdx
dybxy )sin()cos( bxb
dx
dybxy
)(sec)tan( 2 bxbdx
dybxy cxcx ce
dx
dyey
xdx
dyxy
1log
)cosh()sinh( bxbdx
dybxy )sinh()cosh( bxb
dx
dybxy
)(sin
)cos()(cos
2 bx
xb
dx
dybxecy
)(cos)cot( 2 bxecbdx
dybxy
)(cos
)sin()sec(
2 bx
xb
dx
dybxy
NUMERICAL EXAMPLES
KEY TECHNIQUES
1. Define all functions used in the product and quotient rules, with their associated derivatives, clearly.
2. Ensure the layout of the work is uncluttered and unambiguous. This will avoid many typos.
3. Use known results from a table wherever possible.
Slides by Anthony Rossiter
4
Example 1
Find the derivative of:
Slides by Anthony Rossiter
5
)(
)(
)14(
)2log()(
5
32
xv
xu
xx
xxxfy
dx
dpt
dx
dtp
dx
du
xtxpxxu
);()()log32(log2
2v
dx
dvu
dx
duv
dx
dy
Here u(x) is a product of two functions, so we
need the product rule to differentiate this.
xdx
dtx
dx
dp
xxtxxp
3;2
);log32(log)(;)( 2
xxxx
dx
du 3)log32(log2 2
Example 1 - continued
Find the derivative of:
Slides by Anthony Rossiter
6
)(
)(
)14(
)2log()(
5
32
xv
xu
xx
xxxfy
xxxx
dx
du 3)log32(log2 2
dx
dv
xxv ;145
2v
dx
dvu
dx
duv
dx
dy
Substitute into quotient formulae
Example 2
Find the derivative of:
Slides by Anthony Rossiter
7
)(
)(
)3sec(
4)(
2
2
wv
wu
ww
ewgh
w
dw
dpt
dw
dtp
dw
dv
wtwpwwv
);()()3sec(2
2v
dw
dvu
dw
duv
dw
dh
Here v(w) is a product of two functions, so we
need the product rule to differentiate this.
)3(cos
)3sin(3;2
);3sec()(;)(
2
2
w
w
dw
dtw
dw
dp
wwtwwp
Straight from the table of
known results.
Example 2 - continued
Find the derivative of:
Slides by Anthony Rossiter
8
)(
)(
)3sec(
4)(
2
2
wv
wu
ww
ewgh
w
2v
dw
dvu
dw
duv
dw
dh
From previous page.);3sec(2
)3(cos
)3sin(32
2 www
ww
dw
dv
;8;4 22 ww edw
dueu
Straight from the table of known results.
Substitute into quotient formulae
Example 3Find the derivative of:
Slides by Anthony Rossiter
9
)(
)(
)2.0tan(
)3log()5.0cos()5.0sin(6)(
wv
wu
we
wwwwgh
w
2v
dw
dvu
dw
duv
dw
dh
Here u(w) is a product of three functions and v(w) is a product of two functions, so we need the product rule for both.
Next, use product rule to find
derivatives of u(w) and v(w).
However, using tables of known results, students will see a possible double angle formulae in the numerator which will simplify the
overall function.
)(
)(
)2.0tan(
)3log()sin(3)(
wv
wu
we
wwwgh
w
Example 3
Slides by Anthony Rossiter
10
dw
dpt
dw
dtp
dw
du
wtwpwwu
);()()3log()sin(3
Find derivatives of u(w) and v(w) using the product rule.
wdw
dtw
dw
dp
wwtwwp
1);cos(2
);3log()();sin(3)(
)(
)(
)2.0tan(
)3log()sin(3)(
wv
wu
we
wwwgh
w
dw
dqr
dw
drq
dw
dv
wrwqwev w
);()()2.0tan(
)2.0(sec2.0;
);2.0tan()(;)(
2 wdw
dre
dw
dq
wwrewq
w
w
Straight from the table of known results.
Example 3
Slides by Anthony Rossiter
11
Using results of previous page.
)3log()cos(21
)sin(3 www
wdw
du
)(
)(
)2.0tan(
)3log()sin(3)(
wv
wu
we
wwwgh
w
ww ewwedw
dv )2.0tan()2.0(sec2.0 2
2v
dw
dvu
dw
duv
dw
dh
Finally, substitute
into quotient formulae.
Summary
• This video has demonstrated the differentiation of commonplace functions using a lookup table in combination with the product and quotient rules.
• Viewers will see that the most important points are:
– Keep clear definitions of functions used in the product and quotient rules and their derivatives before substituting into the formulae.
– Use a lookup table for common results.
– Don’t worry if the algebra gets messy, but make sure the layout is clear and well organised.
Slides by Anthony Rossiter
12
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Anthony RossiterDepartment of Automatic Control and
Systems EngineeringUniversity of Sheffieldwww.shef.ac.uk/acse