No calculator implicit differentiation: trigonometry

Question

Could you explain how to solve this to someone who has never learned this before? I'm having trouble explaining this to my younger sister.

Find dy/dx at the point (pi, pi/4) when sin x + sin y = tan x + (sqrt2)/2.

A. -2√2

B. 0

C. √(2)/2

D. 2√2

Answer 1

Let's differentiate both sides of sin x + sin y = tan x + (sqrt2)/2.

By the chain rule, d/dx sin y = cos(y)*dy/dx

so d/dx (sin x + sin y ) = d/dx (tan x + (sqrt2)/2)

and therefore

cos(x)+cos(y)*dy/dx = sec^2(x)

so dy/dx at any point (x,y) satisfies this equation.

plugging in x = pi, y = pi/4, we get

cos(pi)+cos(pi)/4 * dy/dx = 1/cos^2(pi)

-1+1/sqrt(2) * dy/dx = 1/(-1)^2

now we can use algebra to solve for dy/dx, or we can just plug in the different multiple choice questions to find which one makes the equation true.