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
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    (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)

    dy/dx = sin(πx)sec(πy)  Answer//

  • 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.

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