Math, asked by aisha723, 14 days ago

cos(x + y) = sin(x - y) implicit differentiation

Answers

Answered by s6d015273shivangisha
1

Answer:

Explanation:

Method 1: Expand and use the product rule

We expand using the formula

cos

(

A

+

B

)

=

cos

A

cos

B

sin

A

sin

B

.

y

=

cos

x

cos

y

sin

x

sin

y

d

d

x

(

y

)

=

d

d

x

(

cos

x

cos

y

sin

x

sin

y

)

d

d

x

(

y

)

=

d

d

x

(

cos

x

cos

y

)

d

d

x

(

sin

x

sin

y

)

Remember that we are differentiating with respect to

x

here.

1

(

d

y

d

x

)

=

sin

x

(

cos

y

)

sin

y

(

d

y

d

x

)

(

cos

x

)

(

cos

x

(

sin

y

)

+

cos

y

(

sin

x

)

d

y

d

x

)

Solve for

d

y

d

x

:

1

(

d

y

d

x

)

+

sin

y

cos

x

(

d

y

d

x

)

+

cos

y

sin

x

(

d

y

d

x

)

=

sin

x

cos

y

cos

x

sin

y

d

y

d

x

(

1

+

sin

y

cos

x

+

cos

y

sin

x

)

=

(

sin

x

cos

y

+

cos

x

sin

y

)

d

y

d

x

=

sin

x

cos

y

+

cos

x

sin

y

1

+

sin

y

cos

x

+

cos

y

sin

x

d

y

d

x

=

sin

(

x

+

y

)

1

+

sin

(

x

+

y

)

Method 2: Use the chain rule

Step-by-step explanation:

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Answered by antrikshbaskotra3036
0

Answer:

d

y

d

x

=

sin

(

x

+

y

)

1

+

sin

(

x

+

y

)

Explanation:

Method 1: Expand and use the product rule

We expand using the formula

cos

(

A

+

B

)

=

cos

A

cos

B

sin

A

sin

B

.

y

=

cos

x

cos

y

sin

x

sin

y

d

d

x

(

y

)

=

d

d

x

(

cos

x

cos

y

sin

x

sin

y

)

d

d

x

(

y

)

=

d

d

x

(

cos

x

cos

y

)

d

d

x

(

sin

x

sin

y

)

Remember that we are differentiating with respect to

x

here.

1

(

d

y

d

x

)

=

sin

x

(

cos

y

)

sin

y

(

d

y

d

x

)

(

cos

x

)

(

cos

x

(

sin

y

)

+

cos

y

(

sin

x

)

d

y

d

x

)

Solve for

d

y

d

x

:

1

(

d

y

d

x

)

+

sin

y

cos

x

(

d

y

d

x

)

+

cos

y

sin

x

(

d

y

d

x

)

=

sin

x

cos

y

cos

x

sin

y

d

y

d

x

(

1

+

sin

y

cos

x

+

cos

y

sin

x

)

=

(

sin

x

cos

y

+

cos

x

sin

y

)

d

y

d

x

=

sin

x

cos

y

+

cos

x

sin

y

1

+

sin

y

cos

x

+

cos

y

sin

x

d

y

d

x

=

sin

(

x

+

y

)

1

+

sin

(

x

+

y

)

Step-by-step explanation:

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