Math, asked by Devanshftp2018, 11 months ago

Differentation of sin inverse x.

Answers

Answered by showdownbegans
1

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:

Answered by saanvi83
2

Answer:

sin inverse of x

Step-by-step explanation:

differentiate with respect to x

dy/dx=cos inverse of x

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