James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.5 Implicit Differentiation 213
If we now substitute Equation 3 into this expression, we get
3x 2 y 3 2 3x 3 y 2S2 x 3
y99 − 2
y 6
y 3D
− 2 3sx 2 y 4 1 x 6 d
y 7
− 2 3x 2 sy 4 1 x 4 d
y 7
But the values of x and y must satisfy the original equation x 4 1 y 4 − 16. So the
answer simplifies to
y99 − 2 3x 2 s16d
− 248 x 2
y 7 y 7
■
Figure 10 shows the graph of the curve
x 4 1 y 4 − 16 of Example 4. Notice
that it’s a stretched and flat tened version
of the circle x 2 1 y 2 − 4. For this
reason it’s sometimes called a fat circle.
It starts out very steep on the left but
quickly becomes very flat. This can be
seen from the expression
y9 − 2 x 3
y 3
S yD − 2 x 3
FIGURE 10
y
2
0
x$+y$=16
2
x
Derivatives of Inverse Trigonometric Functions
The inverse trigonometric functions were reviewed in Section 1.5. We discussed their
continuity in Section 2.5 and their asymptotes in Section 2.6. Here we use implicit
differentia tion to find the derivatives of the inverse trigonometric functions, assuming
that these functions are differentiable. [In fact, if f is any one-to-one differentiable function,
it can be proved that its inverse function f 21 is also differentiable, except where its
tangents are vertical. This is plausible because the graph of a differentiable function has
no corner or kink and so if we reflect it about y − x, the graph of its inverse function also
has no corner or kink.]
Recall the definition of the arcsine function:
y − sin 21 x means sin y − x and 2 2 < y < 2
Differentiating sin y − x implicitly with respect to x, we obtain
cos y dy
dx − 1 or dy
dx − 1
cos y
Now cos y > 0, since 2y2 < y < y2, so
cos y − s1 2 sin 2 y − s1 2 x 2
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