82 STEP 4. Review the Knowledge You Need to Score High
Example 3
Ify=
(
2 x− 1
x^2
) 3
, find
dy
dx
.
Using the chain rule, letu=
(
2 x− 1
x^2
)
. Then
dy
dx
= 3
(
2 x− 1
x^2
) 2
d
dx
(
2 x− 1
x^2
)
.
To find
d
dx
(
2 x− 1
x^2
)
, use the quotient rule.
Thus,
d
dx
(
2 x− 1
x^2
)
=
(2)(x^2 )−(2x)(2x−1)
(x^2 )^2
=
− 2 x^2 + 2 x
x^4
. Substituting this
quantity back into
dy
dx
= 3
(
2 x− 1
x^2
) 2
d
dx
(
2 x− 1
x^2
)
= 3
(
2 x− 1
x^2
) 2
− 2 x^2 + 2 x
x^4
=
−6(x−1)(2x−1)
x^7
2
.
An alternate solution is to use the product rule and rewrite y =
(
2 x− 1
x^2
) 3
as y =
(2x−1)^3
(x^2 )^3
=
(2x−1)^3
x^6
and use the quotient rule. Another approach is to express y =
(2x−1)^3 (x−^6 ) and use the product rule. Of course, you can always use your calculator if
you are permitted to do so.
6.2 Derivatives of Trigonometric, Inverse Trigonometric,
Exponential, and Logarithmic Functions
Main Concepts:Derivatives of Trigonometric Functions, Derivatives of Inverse
Trigonometric Functions, Derivatives of Exponential and
Logarithmic Functions
Derivatives of Trigonometric Functions
Summary of Derivatives of Trigonometric Functions
d
dx
(sinx)=cosx
d
dx
(cosx)=−sinx
d
dx
(tanx)=sec^2 x
d
dx
(cotx)=−csc^2 x
d
dx
(secx)=secxtanx
d
dx
(cscx)=−cscxcotx
Note that the derivatives ofcosine,cotangent, andcosecantall have a negative sign.
Example 1
Ify= 6 x^2 +3 secx, find
dy
dx
.
dy
dx
= 12 x+3 secxtanx.