5 Steps to a 5 AP Calculus AB 2019 - William Ma

(Marvins-Underground-K-12) #1
MA 3972-MA-Book May 9, 2018 10:9

118 STEP 4. Review the Knowledge You Need to Score High


Example 8
Ify=
tanx
1 +tanx
, find
dy
dx

.


Using the quotient rule, letu=tanxandv=(1+tanx). Then,
dy
dx

=


(sec^2 x)(1+tanx)−(sec^2 x)(tanx)
(1+tanx)^2

=


sec^2 x+(sec^2 x)(tanx)−(sec^2 x)(tanx)
(1+tanx)^2

=


sec^2 x
(1+tanx)^2
, which is equivalent to

1


(cosx)^2

1 +

(
sinx
cosx

) 2

=


1


(cosx)^2
(
cosx+sinx
cosx

) 2 =

1


(cosx+sinx)^2

.


Note that for all of the above exercises, you can find the derivatives by using a calculator,
provided that you are permitted to do so.

Derivatives of Inverse Trigonometric Functions
Summary of Derivatives of Inverse Trigonometric Functions
Letube a differentiable function ofx, then
d
dx
sin−^1 u=

1



1 −u^2

du
dx
, |u|< 1
d
dx
cos−^1 u=

− 1



1 −u^2

du
dx
,|u|< 1

d
dx
tan−^1 u=

1


1 +u^2

du
dx

d
dx
cot−^1 u=

− 1


1 +u^2

du
dx
d
dx
sec−^1 u=

1


|u|


u^2 − 1

du
dx
, |u|> 1
d
dx
csc−^1 u=

− 1


|u|


u^2 − 1

du
dx
, |u|> 1.

Note that the derivatives of cos−^1 x, cot−^1 x, and csc−^1 xall have a “−1” in their numerators.

Example 1
Ify=5 sin−^1 (3x), find
dy
dx

.


Letu= 3 x. Then
dy
dx

=(5)


1



1 −(3x)^2

du
dx

=


5



1 −(3x)^2

(3)=


15



1 − 9 x^2

.


Or using a calculator, enterd[5 sin−^1 (3x), x] and obtain the same result.
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