Question

Is this graph increasing or decreasing near (0,pi)
sin(x^2+y)=x

Answer 1

decreasing

Explanation:

You want to know if the graph of sin(x² +y) = x is increasing or decreasing at (x, y) = (0, π).

Graph

An easy way to tell is to look at the graph produced by a graphing calculator. The slope at (0, π) is negative. The graph is decreasing.

Derivative

The derivative of the relation is ...

(2x·dx +dy)(cos(x² +y)) = dx

Dividing by dx and solving for dy/dx gives ...

(2x +dy/dx)cos(x² +y) = 1

(2x +dy/dx) = 1/cos(x² +y)

dy/dx = 1/cos(x² +y) -2x

At the point of interest, this is ...

dy/dx = 1/cos(0 +π) -2·0 = -1

The slope is negative, so the graph is decreasing at (0, π).