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

MA 3972-MA-Book May 9, 2018 10:9

Differentiation 123

Step 3: Factor out
dy
dx

:

dy

dx

[2(x+y)+2(x−y)− 5 y^4 ]= 5 x^4 − 2 x− 2 y+ 2 x− 2 y

dy

dx

[2x+ 2 y+ 2 x− 2 y− 5 y^4 ]= 5 x^4 − 4 y

dy

dx

[4x− 5 y^4 ]= 5 x^4 − 4 y.

Step 4: Solve for
dy
dx

:

dy

dx

=

5 x^4 − 4 y

4 x− 5 y^4

.

Example 4

Write an equation of the tangent to the curvex^2 +y^2 + 19 = 2 x+ 12 yat (4, 3).
The slope of the tangent to the curve at (4, 3) is equivalent to the derivative
dy
dx
at (4, 3).

Using implicit differentiation, you have:

2 x+ 2 y

dy

dx

= 2 + 12

dy

dx

2 y
dy
dx

− 12

dy

dx

= 2 − 2 x

dy

dx

(2y−12)= 2 − 2 x

dy

dx

=

2 − 2 x

2 y− 12

=

1 −x

y− 6

and

dy

dx

∣∣

∣

∣(4, 3)=

1 − 4

3 − 6

= 1.

Thus, the equation of the tangent isy− 3 =(1)(x−4) ory− 3 =x−4.

Example 5

Find
dy
dx
, if sin(x+y)= 2 x.

cos(x+y)

(

1 +

dy

dx

)

= 2

1 +

dy

dx

=

2

cos(x+y)

dy

dx

=

2

cos(x+y)

− 1