Given that y = sin(x+y),find the derivative when (x, y)=(π,0)

Shown in the explanation

Recall that an implicit function is a relation given by the form:

Where is a function of two or more variables. In this case, that function is:

and is implicit because we can define it as:

having two variables.

So, let's take the derivative:

Applying chain rule:

But:

Therefore:

Isolating :

When :

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is