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Laws of logs tells us that ln(a) - ln(b) = ln(a/b)

Therefore ln(10)-ln(5) = ln(10/5) = ln(2)

when differentiating, we multiply the power by the coefficient and then take the power away by 1 so : y= 4x^3 + 3x^2 -5x +1 then 4 x 3= 12 and 3-1 = 2 meaning the solution is 12x^2. 2 x 3 = 6 and 2-1 = 1 ...

Taking the log of both sides we get 5x * ln2 = (2x+1) * ln3.
Taking everything that contains x to the left side: x * (5ln2 - 2ln3) = ln3.
Therefore x=ln3/(5ln2 - 2ln3)
...

3x²+5y+5x(dy/dx)-4y(dy/dx)=0

3x²+5y=(5x-4y)dy/dx

dy/dx=(3x²+5y)/(5x-4y)

Using the quotient rule: Let u=x and v=2x+5 then du/dx= 1 and dv/dx=2 Hence dy/dx=[(2x+5)x1- (2x)]/(2x+5)²             =5/(2x+5)²