Answer:
dy/dx = (sin πx)/(cos πy)
Step-by-step explanation:
(cos πx + sin πy)^(7) = 33
Let's differentiate;
d((cos πx + sin πy)^(7))/dx = d(33)/dx
Now, chain rule is;
d(g(h(x, y)))/dx = = (dg/dh) * (dh/dx)
Let h(x, y) = (cos πx + sin πy)
Then;
dh/dx = [-π sin πx + (πcos πy)*dy/dx]
g = h(x, y)^(7)
dg/dh = 7h(x, y)•d((cos πx + sin πy)^(7))/dx = 7h(x, y)•d((cos πx + sin πy)^(7))/dx•[-π sin πx + (πcos πy)*dy/dx]
This gives;
dg/dh = 7(cos πx + sin πy) • [-π sin πx + (πcos πy)*dy/dx]
The right side is just derivative of a constant: so; d(33)/dx = 0
Thus;
7(cos πx + sin πy) • [-π sin πx + (πcos πy)*dy/dx] = 0
Let's solve for dy/dx by opening the brackets;
Divide both sides by 7(cos πx + sin πy) to get;
[-π sin πx + (πcos πy)*dy/dx] = 0
(πcos πy)*dy/dx = π sin πx
dy/dx = (π sin πx)/(πcos πy)
dy/dx = (sin πx)/(cos πy)
