Jenny asked in Science & MathematicsMathematics · 5 months ago

# Find dy/dx by implicit differentiation. (cos(pi x)+sin(pi y))^7=61?

Find dy/dx by implicit differentiation. (cos(pi x)+sin(pi y))^7=61

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• 5 months ago

(cos(πx)+sin(πy))^7=61

7(cos(πx) + sin(πy)^6 * [-πsin(πx) + πcos(πy)dy/dx] = 0

- 7πsin(πx)(cos(πx) + sin(πy))^6 + 7 πcos(πy)(cos(πx) + sin(πy))^6dy/dx = 0

7 πcos(πy)(cos(πx) + sin(πy))^6dy/dx =  7πsin(πx)(cos(πx) + sin(πy))^6

.............7πsin(πx)(cos(πx) + sin(πy))^6

dy/dx =-----------------------------------------------

..............7 πcos(πy)(cos(πx) + sin(πy))^6

..............sin(πx)

dy/dx = --------------

..............cos(πy)

• 5 months ago

7 * (cos(pi * x) + sin(pi * y))^6 * (cos(pi * y) * pi * dy - sin(pi * x) * pi * dx) = 0

We can write (cos(pi * x) + sin(pi * y))^6 as 61^(6/7)

7 * 61^(6/7) * pi * (cos(pi * y) * dy - sin(pi * x) * dx) = 0

cos(pi * y) * dy - sin(pi * x) * dx = 0

cos(pi * y) * dy = sin(pi * x) * dx

dy/dx = sin(pi * x) / cos(pi * y)

There you go.