derivative of cosec x^2 by first principle?

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  • Philip
    Lv 6
    4 weeks ago

    [csc(x)]^2 = s^(-2), where [s,c] = [sin(x), cos(x)].

    {d/dx}[s^(-2)] = -2s^(-3)*(ds/dx) = -2cs^(-3).

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  • MyRank
    Lv 6
    4 weeks ago

    cosec²x

    d/dx(cosec²x)

    = -2cosecx.cosecx.cotx

    = -2cosec²xcotx

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  • If that's csc(x)^2, then it's just a bit of algebra to find the derivative.  If it's csc(x^2), then that's a horse of a different color.

    f(x) = csc(x)^2

    f(x + h) = csc(x + h)^2

    (f(x + h) - f(x)) / (x + h - x) =>

    (csc(x + h)^2 - csc(x)^2) / h =>

    (1/sin(x + h)^2 - 1/sin(x)^2) / h =>

    ((sin(x)^2 - sin(x + h)^2) / (sin(x + h)^2 * sin(x)^2)) / h =>

    (sin(x)^2 - sin(x + h)^2) / (h * sin(x)^2 * sin(x + h)^2) =>

    ((1/2) * (1 - cos(2x)) - (1/2) * (1 - cos(2x + 2h))) / (h * sin(x)^2 * sin(x + h)^2) =>

    (1/2) * (1 - cos(2x) - 1 + cos(2x + 2h)) / (h * sin(x)^2 * sin(x + h)^2) =>

    (cos(2x + 2h) - cos(2x)) / (2h * sin(x)^2 * sin(x + h)^2) =>

    (cos(2x)cos(2h) - sin(2x)sin(2h) - cos(2x)) / (2h * sin(x)^2 * sin(x + h)^2) =>

    (cos(2x) * (cos(2h) - 1) - sin(2x) * 2sin(h)cos(h)) / (2h * sin(x)^2 * sin(x + h)^2) =>

    (cos(2x) * (cos(h)^2 - sin(h)^2 - 1) - sin(2x) * 2sin(h)cos(h)) / (2h * sin(x)^2 * sin(x + h)^2) =>

    (cos(2x) * (-sin(h)^2 - sin(h)^2) - sin(2x) * 2sin(h)cos(h)) / (2h * sin(x)^2 * sin(x + h)^2) >

    (-2 * sin(h)^2 * cos(2x) - 2 * sin(2x) * sin(h) * cos(h)) / (2h * sin(x)^2 * sin(x + h)^2) =>

    -2 * sin(h) * (sin(h) * cos(2x) + sin(2x)cos(h)) / (2h * sin(x)^2 * sin(x + h)^2) =>

    -sin(h) * (sin(h) * cos(2x) + sin(2x)cos(h)) / (h * sin(x)^2 * sin(x + h)^2)

    An important limit to remember is this:  sin(h)/h goes to 1 as h goes to 0

    -(sin(h)/h) * (sin(h) * cos(2x) + sin(2x) * cos(h)) / (sin(x)^2 * sin(x + h)^2)

    Let h go to 0

    -1 * (0 * cos(2x) + sin(2x) * 1) / (sin(x)^2 * sin(x)^2) =>

    -1 * sin(2x) / sin(x)^4 =>

    -2sin(x)cos(x) / sin(x)^4 =>

    -2cos(x) / sin(x)^3 =>

    -2 * cot(x) * csc(x)^2

    Now, let's use some rules of differentiation to see if this checks out:

    csc(x)^2

    Derive

    2 * csc(x) * (-cot(x) * csc(x)) =>

    -2 * cot(x) * csc(x)^2

    Looks right to me.

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  • alex
    Lv 7
    4 weeks ago

    Is that (cosec x)^2 or cosec (x^2) ?

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