Applications of Differentiation for the TMUA

Gradients, Tangents, and Normals

Differentiation is used to find the rate of change of a curve at a specific point. This rate of change is defined as the gradient of the tangent to the curve at that point. By calculating $\frac{dy}{dx}$, you obtain a function that gives the gradient for any $x$ value. The tangent at a point $(x_1, y_1)$ is a straight line with the same gradient as the curve at that point. The normal is the line perpendicular to the tangent at that point; its gradient $m_n$ satisfies the condition $m_n \times m_t = -1$, where $m_t$ is the gradient of the tangent.

Identifying Stationary Points

Stationary points occur where the tangent to a curve is horizontal, meaning the gradient is zero. To find these points, you must solve the equation $\frac{dy}{dx} = 0$. These points are often referred to as local maxima or local minima because they represent the highest or lowest values of the function in a small local region, though they are not necessarily the global extremes of the function.

Worked Example

Consider the cubic $y = 2x^3 - 3x^2 - 12x + 6$. We can find the $x$-coordinates of its stationary points by differentiating:

$\frac{dy}{dx} = 6x^2 - 6x - 12$

Setting this to zero to find the stationary points:

$6(x^2 - x - 2) = 0$

$6(x - 2)(x + 1) = 0$

This gives stationary points at $x = 2$ and $x = -1$.

Classifying Stationary Points

Once stationary points are located, they must be classified. There are three common techniques:

1. General Shape: For simple polynomials, the leading coefficient often dictates the shape. In the cubic example above, the positive $x^3$ coefficient implies the graph goes from bottom-left to top-right. Thus, the leftmost stationary point ($x = -1$) must be a maximum and the rightmost ($x = 2$) must be a minimum.

2. Neighbouring Values: You can check the $y$ values slightly to the left and right of the stationary point. If the $y$ values on both sides are larger than the value at the point, it is a minimum. If they are smaller, it is a maximum.

3. The Second Derivative Test: This is the most formal method. If $\frac{d^2y}{dx^2} > 0$ at the stationary point, the gradient is increasing, indicating a minimum. If $\frac{d^2y}{dx^2} < 0$, the gradient is decreasing, indicating a maximum.

In the diagram above, the derivative goes from negative to positive, creating a minimum where the second derivative is positive.

In this diagram, the derivative goes from positive to negative, creating a maximum where the second derivative is negative.

Logic of the Second Derivative Test

It is important to understand the logic of these conditions. The condition $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} > 0$ is sufficient but not necessary for a minimum. This is because some minima, such as the origin on $y = x^4$, have $\frac{d^2y}{dx^2} = 0$. Similarly, $\frac{d^2y}{dx^2} < 0$ is a sufficient but not necessary condition for a maximum, as $y = -x^4$ also has a zero second derivative at its maximum.

Strictly Increasing and Decreasing Functions

A strictly increasing function is one where the $y$ value always increases as $x$ increases. This is guaranteed if $f'(x) > 0$ for all $x$ in a given interval. Conversely, a strictly decreasing function always slopes downwards, which is guaranteed if $f'(x) < 0$.

Note that these definitions using derivatives are sufficient conditions. We cannot say that if a function is strictly increasing, its derivative must be positive everywhere. For example, $y = x^3$ is strictly increasing over the whole real line, even though its gradient is zero at the origin. Also, functions with sharp corners can be strictly increasing without having a derivative at every point.

Qualitative Understanding of Points of Inflexion

Points of inflexion are where the curve changes from being 'convex' to 'concave' (or vice versa). While a technical analysis of inflexion points is not required, you should recognise them qualitatively. A classic example is $y = x^3$ at $x = 0$, where the stationary point is neither a maximum nor a minimum but a horizontal point of inflexion.