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To answer this question we require the chain rule, which states that dy/dx=(dy/du)*(du/dx)
To use this formula in our question, we can let y=ln(sin(3x))=ln(u) where u=sin(3x)
Firstly, using a standard result we have dy/du=1/u
Secondly, we must work out du/dx. Another standard result is that d/dx(sin(ax))=a*cos(ax) for any constant number a. This means du/dx=3*cos(3x)
Putting the two parts together, we find that our answer, given by dy/dx, is equal to
There are 2 methods we can use here.
Method 1 - Factorisation
If we can rewrite the equation so that it looks like (x+a)*(x+b)=0, where 'a' and 'b' are numbers, then we can easily find a solution. If one or both of the terms inside the brackets equals 0, then their product is also 0. This means the solution(s) are the values of x such that x+a=0 or x+b=0.
Multiplying out the brackets of (x+a)*(x+b)=0 gives x2+(a+b)*x+a*b=0. If we match up the terms with the equation we want to solve (called comparing coefficients), we see that we want two numbers 'a' and 'b' so that a+b=-5, and a*b=6. The two numbers which satisfy this are -2 and -3, so these are our values for 'a' and 'b'.
Now we can write our equation as (x-2)(x-3)=0 (you can check yourself this can be multiplied out to give the original equation!)
If x-3=0 then x=3, and if x-2=0 then x=2, so these are our two solutions.
Method 2 - Quadratic Formula
This method can be used when you can't easily 'see' the numbers to use to factorise the equation.
The quadratic formula states that for an equation of the form a*x2+b*x+c=0, where 'a', 'b', and 'c' are numbers, then
For our equation a=1, b=-5 and c=6, which can then be substituted into the formula. Again you can check yourself that it simplifies to give the same answers as method 1see more