Ms. Battaglia
AB/BC Calculus
• Up to this point, most functions have been expressed in
explicit form. Ex: y=3x2 – 5
• The variable y is explicitly written as a function of x.
Implicit Form
Explicit Form
Derivative
• How would you find dy/dx for x2 – 2y3 + 4y = 2? You can
use implicit differentiation (apply the Chain Rule, because
you are assuming that y is defined implicitly as a
differentiable function of x).
d é 3ù
ëx û
dx
Variables agree 
d é 3ù
ëy û
dx
Variables disagree 
d
[ x + 3y]
dx
d é 2ù
ë xy û
dx
1.
2.
3.
4.
Differentiate both sides of the equation with
respect to x.
Collect all terms involving dy/dx on the left side
of the equation and move all other terms to the
right side of the equation.
Factor dy/dx out of the left side of the
equation.
Solve for dy/dx.
Find dy/dx given that y3 + y2 – 5y – x2 = -4
If possible, represent y as a differentiable function of x.
a. x2 + y2 = 0
b. x2 + y2 = 1
c. x + y2 = 1
Determine the slope of the tangent line to the
æ
ö
graph of x2 + 4y2 = 4 at the point ç 2, - 1 ÷
è
2ø
Determine the slope of the graph of
3(x2 + y2)2 = 100xy at the point (3,1).
Find dy/dx implicitly for the equation siny=x
2
d
y
2
2
st and
Given x + y = 25, find
.
Evaluate
the
1
dx 2
2nd derivatives at the point (-3,4).
Find the tangent line to the graph given by
x2
(x2
+
y 2)
=
y2
at the point
æ 2 2ö
ç ,
÷.
è 2 2 ø
 Read
2.5, Page 146 #7, 11, 21,
27, 30, 45, 47, 51