Year 2 Differentiation Exam Question
Year 2 Differentiation Exam Question – Introduction
In this article we are going to take a look at a past examination question that is based on a year 2 differentiation exam question and the technique that will be used to perform the differentiation is that of using the quotient rule.
There are two parts to the question as you will see. Part a) relies on the use of differentiation and the second part of the question, part b) relies on using the discriminant.
For both parts of the question it is important that you are able to determine the methods from within the question because at no point does the question give you the method to use. For instance, nowhere in the question are you told to use the quotient rule or to use the discriminant.
You need to be able to recognise the method of differentiation and from the wording of the question what other techniques are required.
The question is as follows:
Dealing with part a) first you need to recognise that f'(x) is telling you to differentiate. You can see that you have f(x) written as a fraction so there is your clue that you need to be using the quotient rule.
Generally u = numerator and v = denominator and you also need to make sure that you know what the rule is for the quotient rule. Which is:
\frac{d y}{d x}=v \frac{d u}{d x}-u \frac{d v}{d x}Working it out
Having identified these variables you can then perform the differentiation.
\begin{aligned} u & =e^{3 x} \\ \frac{d u}{d x} & =3 e^{3 x} \end{aligned}\begin{aligned} v & =4 x^2+k \\ \frac{d v}{d x} & =8 x \end{aligned}
This information can then be substituted into the quotient rule formula which is written above and this will give the following:
\begin{aligned} f^{\prime}(x) & =\frac{3\left(4 x^2+k\right) e^{3 x}-e^{3 x}(8 x)}{\left(4 x^2+k\right)^2} \\ & =\frac{\left(12 x^2+3 k\right) e^{3 x}-8 x e^{3 x}}{\left(4 x^2+k\right)^2} \\ \therefore f^{\prime}(x) & =\frac{\left(12 x^2-8 x+3 k\right) e^{3 x}}{\left(4 x^2+k\right)^2} \end{aligned}
How to tackle this particular Year 2 Differentiation Exam Question
When doing the above you will notice two things:
The main thing to notice is that the denominator has not been expanded. Expanding does not always help, so try to avoid this.
The second thing to notice is that the exponential has always been kept out and in the final working the exponential has been kept out as a factor. This was known to be done using the question and the answer that it has provided as a guide to get me to the desired answer.
The question is asking to determine what g(x) is and this can be written as follows:
g(x)=\frac{e^{3 x}}{\left(4 x^2+k\right)^2}This now brings us to part b) of the question.
It was mentioned above that the discriminant needs to be used. How do we know?
This is because we are told there is at least one stationary point. A stationary point is when the gradient function is equal to zero. So if there is at least one then that means the discriminant must be used and for this particular case b^2-4 a c>0.
A stationary point takes place when the gradient function is equal to zero which means we then have the following:
\left(12 x^2-8 x+3 k\right) e^{3 x}=0\begin{gathered} 12 x^2-8 x+3 k=0 \\ b^2-4 a c>0 \\ (-8)^2-4(12)(3 k)>0 \\ 64>144 k \\ k<\frac{4}{9} \end{gathered}\therefore \quad 0<k<\frac{4}{9}Because we are told in the question that k is a positive constant we have the range of values as shown above and this is our final answer to part b) of the question.