Differentiation Rational Powers

Differentiation of powers of x

Differentiation of $x^n$ for rational $n$, and related sums and differences. This might require some simplification before differentiating — for example, the ability to differentiate an expression such as $\frac{(3x+2)^2}{x^2}$.

In the TMUA/ESAT, we expect you to be able to differentiate simple expressions involving sums of powers of $x$ or expressions that can be simplified to sums of powers of $x$. We do NOT expect you to be able to differentiate trigonometric expressions or use rules like the chain rule, the product rule etc. We have kept to the scope of what we expect you to be able to differentiate both narrow and simple because we want to be able to test your understanding of the topics rather than how good you are using standard algorithms.

We expect you to know the rule for differentiating a power of $x$:

$\frac{\mathrm{d}}{\mathrm{d}x} x^n = n x^{n-1}$

And notice that we use $\frac{\mathrm{d}}{\mathrm{d}x}$ to mean "differentiate this".

And [if you have looked at differentiation from first principles, which is not on the TMUA/ESAT specification!] you should think about why the result when you differentiate $x^n$ is actually $\binom{n}{1}x^{n-1}$ when $n$ is an integer.

In general, if you are not using some of the more advanced rules such as the chain rule or product rule, the best thing to do when differentiating a given expression is to simplify first to make it into a sum of powers of $x$ and then differentiate term by term. In fact, one thing we have not yet mentioned [because it is usually taken as obvious in a first course in calculus] is that you can differentiate term by term and add up the result to get the differential of an expression. (It is usually somewhat dangerous to assume something is obvious in mathematics until you have spent time convincing yourself that it is. Many mistakes in more advanced mathematics can arise when a technique or idea is used incorrectly because it seemed obvious to use it that way: for instance, it might seem "obvious" to write something like $(x + y)^2 = x^2 + y^2$ but it is, in fact, generally mathematical bunkum. Mathematicians sometimes use words such as "trivial" and "obvious", but what they often mean is that they have thought deeply about it, often for some time [even weeks or months or years], and only after all this deep thought is the idea they are referring to "obvious" etc. So do not worry if something that someone says is "obvious" is not so obvious to you!) For instance, you can do this:

$\frac{\mathrm{d}}{\mathrm{d}x}(x^3 + 7x^2 - 3x + 11) = \frac{\mathrm{d}}{\mathrm{d}x}x^3 + \frac{\mathrm{d}}{\mathrm{d}x}7x^2 - \frac{\mathrm{d}}{\mathrm{d}x}3x + \frac{\mathrm{d}}{\mathrm{d}x}11 = 3x^2 + 14x - 3 + 0$