Math

Could I please get some help for these 3 questions, apologies for asking so many, but they were the ones I was really unsure of:

1. Find an equation for the line tangent to y=ln(x) and that goes through the origin.

I have no idea, the ln(x) stuff is confusing me.

2. derivative of y=log_3(1+xln(3))? (the _3 means base 3). I got 1/(ln(3)*(ln(3)+(x/3))/(1+xln(3)), but I don't think i did this right- is the second part of the product rule where we take the derivative of the ln(3) multiply by x supposed to be 0 or x/3? Could you show your work to me too to check?

3. Derivative of ln(x/(√ (x+1)))? I don't know how to simplify further: (√ (x+1))/x * (√ (x+1))/(x+1) - x/[(2(x+1)(√ (x+1))].

asked by Bob
  1. 1.
    y = lnx
    dy/dx = 1/x

    let the point of contact be (a, lna)
    at (a, lna) , the slope is 1/a

    slope using the grade 9 way of doing slope
    = (lna - 0)/(a-0) = lna/a

    but lna/a = 1/a
    lna = 1
    so a = e , (which is appr 2.71828..)

    so we want the equation of a line with slope 1/e and point (e, 1)
    y-1 = (1/e)(x-e)
    fix up any way you need to

    2. y = log3 (1 + xln3)

    dy/dx = (1/ln3) (ln3/(1+xln3)
    = 1/(1 + xln3)

    3. May I suggest you use rules of logs and change
    y = ln (x/√(x+1) ) to
    y = lnx - ln √(x+1)
    = lnx - (1/2)ln(x+1)

    now take the derivative

    posted by Reiny

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