Differentiation of Rational Powers for the TMUA

Differentiation is a fundamental tool in calculus used to determine the rate of change of a function. In the context of the TMUA, you are expected to understand its geometric interpretation as the gradient of a tangent and its algebraic application to polynomial-style expressions.

Derivatives as Rates of Change

A derivative tells you how fast one variable changes relative to another. For example, speed is the rate of change of distance with respect to time. If an object travels at 3 metres per second, its distance increases by 3 units for every 1 unit of time. In a coordinate plane, the gradient of a curve at a point tells us how much $y$ changes for every 1 unit change in $x$. This is interpreted geometrically as the gradient of the tangent to the curve at that specific point.

You must be familiar with the following notation:

1. $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ (Leibnitz notation for the first and second derivatives).
2. $f'(x)$ and $f''(x)$ (Function notation).
3. In mechanics, a dot above a variable (e.g. $\dot{s}$ for speed) indicates differentiation with respect to time, though this is less common in TMUA Paper 1.

The Power Rule for Rational Exponents

The primary rule for differentiation in this specification is the power rule. For any expression in the form $x^n$, where $n$ is a rational number:

$\frac{d}{dx}(x^n) = nx^{n-1}$

When differentiating a sum or difference of terms, you simply differentiate each term individually and add or subtract the results. For example:

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

Note that the derivative of a constant term (like 11) is always 0, as constants do not change with respect to $x$.

Simplification Before Differentiating

Many TMUA questions provide expressions that do not immediately look like a sum of $x^n$ terms. Since advanced rules like the product rule or quotient rule are not required for this specification, you must simplify the expression algebraically first. This typically involves expanding brackets or dividing through by a common denominator.

Worked Example: Differentiating an Algebraic Fraction

Differentiate $y = \frac{(3x+2)^2}{x^2}$ with respect to $x$.

1. First, expand the numerator: $(3x+2)^2 = 9x^2 + 12x + 4$.
2. Rewrite the function by dividing each term in the numerator by the denominator: $y = \frac{9x^2}{x^2} + \frac{12x}{x^2} + \frac{4}{x^2}$.
3. Simplify to index form: $y = 9 + 12x^{-1} + 4x^{-2}$.
4. Now, apply the power rule to each term: $\frac{dy}{dx} = 0 + (-1)(12x^{-2}) + (-2)(4x^{-3})$.
5. Final result: $\frac{dy}{dx} = -12x^{-2} - 8x^{-3}$ or $-\frac{12}{x^2} - \frac{8}{x^3}$.

Stationary Points and Second Derivatives

Stationary points occur where the tangent to a curve is horizontal, meaning the gradient is zero. To find these points, you solve:

$\frac{dy}{dx} = 0$

Once a stationary point is found, you must classify it as a local maximum or a local minimum. The second derivative, $\frac{d^2y}{dx^2}$, measures how the gradient itself is changing:

1. If $\frac{d^2y}{dx^2} > 0$ at the stationary point, the gradient is increasing, signifying a minimum.
2. If $\frac{d^2y}{dx^2} < 0$ at the stationary point, the gradient is decreasing, signifying a maximum.

Note that if $\frac{d^2y}{dx^2} = 0$, the test is inconclusive. The point could be a maximum, a minimum, or a point of inflexion (such as in the case of $y = x^4$ or $y = x^3$).

Increasing and Decreasing Functions

A function is strictly increasing on an interval if its gradient is always positive throughout that interval ($f'(x) > 0$). Conversely, it is strictly decreasing if its gradient is always negative ($f'(x) < 0$).

While points of inflexion will not be examined in detail, you should qualitatively understand that they represent a point where the curve changes its concavity, such as the origin in the graph of $y = x^3$.