Then
where we substituted for from (1) in (2), then used the given equation to
simplify in (3).
Example 31 __
Using implicit differentiation, verify the formula for the derivative of the inverse
sine function, y = sin−^1 x = arcsin x, with domain [−1,1] and range .
SOLUTION: y = sin−^1 x x = sin y.
Now we differentiate with respect to x:
where we chose the positive sign for cos y since cos y is nonnegative if
. Note that this derivative exists only if −1 < x < 1.
Example 32 __
Where is the tangent to the curve 4x^2 + 9y^2 = 36 vertical?
SOLUTION: We differentiate the equation implicitly to get ,
so . Since the tangent line to a curve is vertical when = 0, we
conclude that must equal zero; that is, y must equal zero with x ≠ 0. When
we substitute y = 0 in the original equation, we get x = ±3. The points (±3,0) are
the ends of the major axis of the ellipse, where the tangents are indeed vertical.