Differentiating Logarithmic Functions - Grade 12 Calculus *Help?

Question

i have a few questions in my homework that i could not do and need help on, here they are:

5f) differentiate the function
y=ln(x+1/x-1)

9b) Determine the coordinates of all max or min points of each function
y= x ln x^2
y= x^2 / ln x^2

16) Determine the equation of the tangent to the graph of y= ln x that has slope e

19c) differentiate each function
x ln y =1

d) sin(ln x)


THANKS IN ADVANCE

Answer 1

Remember that the derivative of ln(x) is 1/x, so to take the derivative of ln( f(x) ), you have to apply the chain rule, making its derivative (1/x) * f'(x)

5f) y = ln[(x+1)/(x-1)]
dy/dx = (1 / [(x+1)/(x-1)] ) * [ 1/(x-1) + (x+1)ln(x-1) ]
dy/dx = [ (x-1) / (x+1) ] * [ 1/(x-1) + (x+1)ln(x-1) ]
dy/dx = [1/(x+1)] + (x-1)ln(x-1)

9b) To take the derivative of y= x ln(x^2), treat x and ln(x^2) as separate terms, then take the derivative in the same way you'd derivate the product of two functions:
1 * ln(x^2) + x * (1/(x^2) * 2x) =
ln(x^2) + 2

16) y' (x)= 1/x, so the slope is going to be e when 1/x = e, which is when x = 1/e. To find the corresponding y value at this point, plug 1/e into the equation: y=ln(1/e), y = ln(e^-1),
y = -ln(e) = -1. Now you just have to find a line whose slope is e and which includes the point (1/e, -1). So use y = mx+b and figure out m and b.

19c) Note that you can rewrite this as y = e^(1/x)
d) Again, you have f( g(x) ) here, so find f ' ( g(x) ) * g'(x).

Answer 2

y´= (x-1)/(x+1)* 1/(x-1)^2 *(-2)=-2/(x^2-1)
9b) y´=ln x^2 +x*1/x^2*2x = ln x^2+2 (be careful not to put lnx^2=2lnx because the left side exists for all x not zero while the right side exists only for x>0.
In any case lnx^2 = 2ln IxI
lnx^2=-2 so x^2=e^-2 and x= +- 1/e
Observe that y is an odd function
The other is an even function you should work out
16)
y´=1/x= e so x=1/e
and y =-1
so the equation is y+1=e(x-1/e)
y=e* x-2

19c) lny +x*y´/y=0

y´= cosln(x)*1/x