Составители:
Рубрика:
Example 15
Find the inverse function of f(x)=x
3
+2.
According to the previous Remark we first write y = x
3
+2.Then
we solve this equation for x : x
3
= y − 2, x =
3
√
y − 2.
Finally, we interchange x and y : y =
3
√
x − 2. Therefore, the
inverse function is f
−1
(x)=
3
√
x − 2.
Remark 7
The graph of f
−1
is obtained by reflecting the graph of f about
the line y = x.
Theorem 8
If there is an inverse function x = h(y) for a given function
y = f(x), and if the derivative f
(x) is not zero at a given number x,
then the inverse function h is differentiable at the number y = f(x)
and
h
(y)=
1
f
(x)
.
Example 16
Derivatives of Inverse Trigonometric Functions
Recall that y =sin
−1
x means sin y = x and −π/2 y
π/2. Differentiating sin y = x implicitly with respect to x,weobtain
cos y
dy
dx
=1 or
dy
dx
=
1
cos y
.
Now cos y 0, since −π/2 y π/2,so
cos y =
1 − sin
2
y =
√
1 − x
2
.
Therefore
dy
dx
=
1
cos y
=
1
√
1 − x
2
,
d
dx
(sin
−1
x)=
1
√
1 − x
2
.
The formula for the derivative of the inverse tangent function is
derived in a similar way. If y =tan
−1
x,thentan y = x. Differentiating
25
Страницы
- « первая
- ‹ предыдущая
- …
- 23
- 24
- 25
- 26
- 27
- …
- следующая ›
- последняя »
