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Differentiate with respect to x

  1. Jan 4, 2012 #1
    1. The problem statement, all variables and given/known data

    Differentiate -sin2x

    2. Relevant equations
    CHAIN RULE dy/dx=dz/dx*dy/dz
    PRODUCT RULE

    3. The attempt at a solution

    Using the chain rule;

    let z = -sinx let y = z2

    dz/dx = -cosx dy/dz = 2z

    dy/dx=dz/dx*dy/dz = -cosx*2-sinx = 2cosx*sinx

    If I use the chain rule:

    let u = -sinx let v = -sinx

    du/dx = -cosx dv/dx = -cosx

    udv/dx+vdu/dx = -sinx*-cosx + -sinx*-cosx

    = 2sinx*cosx

    My book answer says -2sinxcosx ?

    Also on a website says:

    Expression

    Function -sin(x)^2
    f'x = -2*cos(x)*sin(x)

    Thanks for any help.
     
  2. jcsd
  3. Jan 4, 2012 #2

    CompuChip

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    (-sin x)2 is not the same as - (sin2 x).
    If you let z = -sin x, then y = -z2.
     
  4. Jan 4, 2012 #3
    What about -sinx*sinx. Does this equal -sinx2 or -sin2x ? or what you have put?

    I would write (-sin x)2 = (-sinx)(-sinx) ? (but this = sinx2 and thus it might be -(sinx*sinx) ?and you are saying it is not equal to - (sin2 x) = -sinx*sinx. I think?

    Could you put a few lines together to explain this please?

    Why did you put the - sign outside of the brackets?

    If I do it like you said I get:

    let z = -sinx let y = -z2

    dz/dx = -cosx dy/dz = -2z

    dy/dx=dz/dx*dy/dz = -cosx*-2-sinx = -2cosx*sinx

    I thought that the minus sign would be taken after within Z and it would not need to be added outside as in the first instance?


    Thanks.
     
  5. Jan 4, 2012 #4

    CompuChip

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    Homework Helper

    Squaring has higher priority than multiplication (by -1), so when we write -z2, we mean -(z2) = - z * z.
    This is something else than (-z)2, which would be (-z) * (-z) = z * z = z2.

    In the case of a sine function, we abbreviate (sin x)2 to sin2 x. This is done to distinguish from sin x2 = sin(x2) while not having to write the brackets.

    So -sin2 x means negative the square of the sine of x: -1 * (sin x) * (sin x), as opposed to -sin(x2) (which cannot be simplified further) or (-sin x)2 = (-sin x) * (-sin x) = sin x * sin x = sin2 x.
     
  6. Jan 4, 2012 #5
    Excellent, many thanks.
     
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