Math 400 (Section 3.7)
Exploration: Implicit Differentiation
Name ____________________________
We have discovered (and proved) formulas for finding derivatives of functions like
f ( x ) = x3 − 3x 2 + 4 x + 2 .
dy
for y = x3 − 3 x 2 + 4 x + 2 . Notice that in this case, y is expressed
dx
explicitly in terms of x. But we need not have y explicitly in terms of x to find y′ .
This amounts to finding
1. Consider the relation x 2 + y 2 = 25 . Graph the relation at
right. (Note that this relation implicitly defines two
functions.)
2. Find the slopes of the lines tangent to the graph where
x = 4 using the fact that any line tangent to a circle is
perpendicular to the radius from the center to the point of
tangency.
Slope1 = ____________
Slope2 = ____________
3. Now let's find these slopes algebraically by using implicit differentiation. Notice that the
relationship between x and y is defined implicitly within the equation. We say this because y
is not alone on one side of the equation. Start with the original equation, and differentiate
both sides of the equation with respect to x. Since y is a function of x, be sure to apply the
Chain Rule (and let y′ denote the derivative of y with respect to x). Solve for y′ in terms of x
and y. Finally, use that fact that (4, 3) and (4, -3) are the points of tangency.
Slope1 = _________
Slope2 = _________
4. Extension of the Power Rule to Rational Exponents.
p
Let y = x q . We might be tempted to use the power rule to find y′ directly here, but so far,
we have only proved that the Power Rule works for integer powers, not for rational powers
in general. Let's use implicit differentiation to see if the Power Rule works for rational
powers. Raise both sides to the q power. Then we have integer exponents, so the Power Rule
applies. Now differentiate implicitly and solve for y′ .
5. The figure at right shows the graph of the
relation x 2 − 4 xy + 4 y 2 − 10 x = −5 . Verify
that the points (3, 4) and (3, -1) lie on the
graph.
6. Find the equations of the lines tangent to the curve at (3, 4) and (3, -1). [Hint: Use implicit
differentiation.] Then graph the lines above to make sure they are reasonable.
Line 1: __________________________________
Line 2: __________________________________
7. Now see if you can solve the equation of the relation given in Problem 5 for y in terms of x.
[Hint: Use the Quadratic Formula.]
8. Which would be easier…Using implicit differentiation, as we did here, or to solve for y
explicitly and then differentiate?