Trig Functions Ms. Roque 11 th Grade Trigonometry Course Next slide.
MATH 31 LESSONS Chapters 6 & 7: Trigonometry 9. Derivatives of Other Trig Functions.
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Transcript of MATH 31 LESSONS Chapters 6 & 7: Trigonometry 9. Derivatives of Other Trig Functions.
MATH 31 LESSONS
Chapters 6 & 7:
Trigonometry
9. Derivatives of Other Trig Functions
Section 7.3: Derivatives of Other
Trigonometric Functions
Read Textbook pp. 315 - 319
Derivatives of Other Trig Functions
We will now look at the derivatives of other
trig functions.
Each of these formulas are derived from the
derivatives of sine / cosine, as will be shown
in the first example.
Formulas:
xxdx
d 2sectan
xxdx
d 2csccot
xxxdx
dcotcsccsc
xxxdx
dtansecsec
Note: If the trig functions have an “inside function u(x)”,
then the formulas can be altered as follows:
dx
duuu
dx
d 2sectan
dx
duuu
dx
d 2csccot
dx
duuuu
dx
d cotcsccsc
dx
duuuu
dx
d tansecsec
Ex. 1 Prove that
Try this example on your own first.Then, check out the solution.
xxdx
d 2csccot
x
x
dx
dx
dx
d
sin
coscot
Reduce the trig function to sine and cosine.
x
x
dx
dx
dx
d
sin
coscot
2sin
sincossincos
x
xxxx
2v
vuvu
v
u
Quotient Rule:
x
x
dx
dx
dx
d
sin
coscot
2sin
sincossincos
x
xxxx
2sin
coscossinsin
x
xxxx
2sin
coscossinsin
x
xxxx
2
22
sin
cossin
x
xx
2
22
sin
cossin
x
xx
Simplify and factor out the negative from the numerator.
2sin
coscossinsin
x
xxxx
2
22
sin
cossin
x
xx
2
22
sin
cossin
x
xx
x2sin
1
x2csc 1cossin 22
Ex. 2 Differentiate.
Try this example on your own first.Then, check out the solution.
3
4
1sec12 xy
3
4
1sec12 xy
3
4
1Let xxuu
uy sec12
If it helps, substitute u(x) for the “inside function”
uy sec12
dx
duuuy tansec12
Find the derivative of the “outside function”.
Leave the inside function the same.
Don’t forget to find the derivative of the “inside function”
dx
duuuy tansec12
3
4
1Let xxuu
333
4
1
4
1tan
4
1sec12 x
dx
dxxy
Back substitute.
333
4
1
4
1tan
4
1sec12 x
dx
dxxy
233 3
4
1
4
1tan
4
1sec12 xxx
332
4
1tan
4
1sec9 xxx
Ex. 3 Differentiate.
Try this example on your own first.Then, check out the solution.
xy costan2
xy costan2
2costan xy
Remember, the entire tangent function is being squared.
i.e. tan 2 x = (tan x)2
xy costan2
2costan xy
xdx
dxy costancostan2
Find the derivative of the “outside function”.
Leave the inside function the same.
Don’t forget to find the derivative of the “inside function”
Chain rule
xdx
dxy costancostan2
xdx
dxx coscosseccostan2 2
Chain rule again.
xdx
dxy costancostan2
xdx
dxx coscosseccostan2 2
xxx sincosseccostan2 2
xxx costancossecsin2 2
Ex. 4 Differentiate.
Try this example on your own first.Then, check out the solution.
12csc
1
xy
2
112csc
12csc
1
xx
y
First, write in exponential notation.
2
112csc
12csc
1
xx
y
12csc12csc2
12
3 x
dx
dxy
Find the derivative of the “outside function”.
Leave the inside function the same.
Don’t forget to find the derivative of the “inside function”
Chain rule
2
112csc
12csc
1
xx
y
12csc12csc2
12
3 x
dx
dxy
1212cot12csc12csc2
12
3 x
dx
dxxx
Chain rule again.
1212cot12csc12csc
2
12
3 x
dx
dxxx
212cot12csc12csc2
12
3 xxx
1212cot12csc12csc
2
12
3 x
dx
dxxx
212cot12csc12csc2
12
3 xxx
12cot12csc12csc22
12
3 xxx
Bring all the coefficients together
1212cot12csc12csc
2
12
3 x
dx
dxxx
212cot12csc12csc2
12
3 xxx
12cot12csc12csc 23
xxx
12cot12csc12csc22
12
3 xxx
12cot12csc12csc 23
xxx
12cot12csc 21
xx
Notice,
u3/2 u1 = u1/2
12cot12csc12csc 23
xxx
12cot12csc 21
xx
12csc
12cot
x
x