calculus

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find dy/dx

1. y=(lnx)^3

2. y=x^4lnx

3. y=sin^3(lnx)

  • calculus -

    1. use chain rule
    u = lnx
    y = u^3
    dy/dx = 3u^2 du/dx = 3(lnx)^2/x

    2. use product rule
    f=x^4
    g=lnx
    y = fg
    y' = f'g + fg' = 4x^3 lnx + x^4/x = x^3(4lnx + 1)

    3. use chain rule
    u = lnx
    v = sin(u)
    y = v^3

    dy/dx = dy/dv dv/du du/dx
    = 3v^2 * cos(u) * 1/x
    = 3sin^2(lnx) * cos(lnx) * 1/x

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