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

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


Example 1

Find
dy
dx

ify^2 − 7 y+x^2 − 4 x=10.

Step 1: Differentiate each term of the equation with respect tox. (Note thatyis treated as
a function ofx.) 2y
dy
dx

− 7


dy
dx
+ 2 x− 4 = 0

Step 2: Move all terms containing
dy
dx
to the left side of the equation and all other terms

to the right side: 2y
dy
dx

− 7


dy
dx
=− 2 x+4.

Step 3: Factor out
dy
dx

:


dy
dx
(2y−7)=− 2 x+ 4.

Step 4: Solve for
dy
dx

:


dy
dx

=


− 2 x+ 4
(2y−7)

.


Example 2

Find
dy
dx

ifx^3 +y^3 = 6 xy.

Step 1: Differentiate each term with respect tox:3x^2 + 3 y^2
dy
dx
=(6)y+

(
dy
dx

)
(6x).

Step 2: Move all
dy
dx
terms to the left side: 3y^2
dy
dx
− 6 x
dy
dx
= 6 y− 3 x^2.

Step 3: Factor out
dy
dx

:


dy
dx
(3y^2 − 6 x)= 6 y− 3 x^2.

Step 4: Solve for
dy
dx

:


dy
dx

=


6 y− 3 x^2
3 y^2 − 6 x

=


2 y−x^2
y^2 − 2 x

.


Example 3

Find
dy
dx

if (x+y)^2 −(x−y)^2 =x^5 +y^5.

Step 1: Differentiate each term with respect tox:

2(x+y)

(
1 +
dy
dx

)
−2(x−y)

(
1 −
dy
dx

)
= 5 x^4 + 5 y^4
dy
dx

.


Distributing 2(x+y) and −2(x−y), you have

2(x+y)+2(x+y)
dy
dx
−2(x−y)+2(x−y)
dy
dx
= 5 x^4 + 5 y^4
dy
dx

.


Step 2: Move all
dy
dx
terms to the left side:

2(x+y)
dy
dx
+2(x−y)
dy
dx
− 5 y^4
dy
dx
= 5 x^4 −2(x+y)+2(x−y).
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