Solve the equation d/dx((x^3 + 3x^2)ln(x)) = 2x^2 + 5x, leaving your answer as an exact value of x. [6 marks]
This may look complicated at first but it can be broken down into a number of simple steps...
For the left side of the equation, the d/dx tells you we are differentiating and since (x3 + 3x2)ln(x) sees two functions multiplied together, we therefore use the product rule.
The product rule looks like this:
u v' + u' v
where in this case u = (x3 + 3x2) and v = ln(x)
u' is the differential of u. Differentiating here simply involves multiplying the coefficient (number in front of the x) by the power and then subtracting one from the power so:
u' = 3x2 + 6x
v' is the differential of v so:
v' = 1/x
Now we simply put the values into our product rule equation and put into our product rule equation:
u v' + u v'
(x3 + 3x2)(1/x) + (3x2 + 6x)(ln(x))
Expanding the first set of brackets gives:
x2 + 3x + (3x2 + 6x)(ln(x))
Putting this into the equation stated in the question gives:
x2 + 3x + (3x2 + 6x)(ln(x)) = 2x2 + 5x
Rearrange and simplify as follows:
(3x2 + 6x)(ln(x)) = x2 + 2x
ln(x) = (x2 + 2x) / (3x2 + 6x)
Now if we factorise the bottom of the fraction:
ln(x) = (x2 + 2x) / (2(x2 + 3x))
And cancelling the x2 + 2x terms:
ln(x) = 1/3
Solving for x gives:
x = e(1/3)
