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

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).