Differentation of sin inverse x.
Answers
Answer:
1. Implicit differentiation
In math 1, we learned about the function ln x being the inverse of the function
e
x
. Remember that we found the derivative of ln x by differentiating the equation
ln x = y.
First, you wrote it in terms of functions that we knew:
x = e
y
Then, we took the derivative of both sides
1 = e
y
dy
dx.
Then, since e
y = x, we simplified to
1 = x
dy
dx
and concluded by dividing both sides by x to get
1
x
=
dy
dx.
2. Inverse trig functions
We will do the same for the inverse trig functions. The process is the same, it is
just a little hard to simplify.
Example 1. Find the dy
dx when y = Sin−1
(x).
Solution. Again we start by writing it in terms of functions we know better, so
sin(y) = x
for y ∈
−π
2
,
π
2
. Now, take the derivative of both sides,
cos(y)
dy
dx = 1.
Now y = Sin−1
(x) so we need to simplify cos(Sin−1
(x)). We did this in Example 5
of the previous packet where we showed
cos(Sin−1
(x)) =
p
1 − x
2.
So we conclude that
dy
dx =
1
√
1 − x
2
.
1
2 DERIVATIVES
Patterning our work after the example we can show that
(1) for y = Tan−1
(x), we get
dy
dx =
1
1 + x
2
(2) for y = Cos−1
(x), we get
dy
dx =
−1
√
1 − x2
3. Problems
Repeat the Example for
(1) y = Tan−1
(x)
(2) y = Cos−1
(x)
Find the derivatives of the following functions
(1) f(x) = Sin−1
(2x − 1).
(2) h(x) = (1 + x
2
)Tan−1
(x).
(3) y = cos−1
t
t
.
(4) g(x) = Tan−1
(sin(x)).
(5) y = Tan−1
x
a
+ ln qx−a
x+a
.
(6) F(t) =
√
1 − t
2 + Sin−1
t.
(7) f(x) = x sin xCos−1
x
(8) y = (Sin−1
x)
2
(9) y = Sin−1
x
2
(10) U(t) = e
Tan−1
t
.
Solutions to the odd numbered ones of the last 10:
• (1) √
2
1−(2x−1)2
• (3)
√−t
1−t2
−cos−1
t
t
2
• (5)
1
a
1+(
x
a )
2 +
1
2 (
x−a
x+a )
−1
2
(x+a)−(x−a)
(x+a)2
qx−a
x+a
• (7) sin x cos−1 x + x
cos x cos−1 x − √
sin x
1−x2
• (9) √
2x
1−x4
Step-by-step explanation:
Answer:
sin inverse of x
Step-by-step explanation:
differentiate with respect to x
dy/dx=cos inverse of x