A-level Mathematics/CIE/Pure Mathematics 1/Differentiation

The Gradient of a Point on a Curve


The gradient of a point on a curve is equal to the gradient of the tangent line at that point. Since it is difficult to measure the gradient of the tangent directly, we use secant lines which approach the tangent. A secant line is a line that travels between two points on the curve.

The Formal Definition of a Derivative
Suppose we want to find the gradient of a point at coordinates $$(x,y)$$. We can approximate it using a secant line that travels through $$(x,y)$$ and $$(x+\Delta x,y+\Delta y)$$ where $$\Delta x$$ is a small change in $$x$$ and$$\Delta y$$ is the small change in $$y$$ that results from it.

The gradient of the secant line is $$\frac{rise}{run} = \frac{y+\Delta y - y}{x+\Delta x - x} = \frac{\Delta y}{\Delta x}$$. As $$\Delta x$$ gets smaller and smaller, $$\frac{\Delta y}{\Delta x}$$ approaches the gradient of the point $$(x,y)$$.

The limit of $$\frac{\Delta y}{\Delta x}$$ as $$\Delta x$$ approaches zero is the value that we are looking for: the gradient of the point $$(x,y)$$. For an arbitrary point $$(x,y)$$, the gradient can be given by the derivative.

The derivative $$f'$$ of a function $$f$$ is a function which provides the gradient of a point $$(x,f(x))$$ on the curve produced by $$f$$. The derivative is formally defined as $$\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}$$

Notation
There are two main ways to write the derivative of a function: $$f'(x)$$ and $$\dfrac{df}{dx}$$. These both mean the same thing: the derivative of $$f$$ with respect to $$x$$.

Simple Rules
These are a few simple rules that make differentiation of complicated expressions easier.

The Power Rule
$$\dfrac{d}{dx} x^n = nx^{n-1}$$

e.g. $$\dfrac{d}{dx} x^3 = 3x^{2}$$

The Sum Rule
$$\dfrac{d}{dx} \big(f(x) + g(x)\big) = f'(x) + g'(x)$$

e.g. $$\dfrac{d}{dx} (x^2 + x^4) = 2x + 4x^3$$

The Constant Rule
$$\dfrac{d}{dx} \big(kf(x)\big) = kf'(x)$$

e.g. $$\dfrac{d}{dx} (4x^2) = 4(2x) = 8x $$

The Chain Rule
$$\dfrac{d}{dx} f(g(x)) = f'(g(x))\times g'(x)$$

e.g. $$\dfrac{d}{dx} (2x+1)^3 = 3(2x+1)^2 \times 2 = 6(2x+1)^2$$

Tangents & Normals
A tangent is a line which travels through a given point and has a gradient which is equal to the gradient of the curve at that point. A normal is a line which travels through a given point and is perpendicular to the tangent of the curve at that point.

To find the equation of the tangent or normal of a curve at a particular point, we can use the equations of a line that we used in ../Coordinate Geometry/.

e.g. Find the equation of the tangent and normal of a curve $$x^3 - 5x^2 + 2x + 6$$ at the point $$(2,-2)$$

$$\begin{align} \dfrac{d}{dx} (x^3 - 5x^2 + 2x + 6) &= 3x^2 - 10x + 2 \\ \text{Gradient at }(2,-2) = 3(2)^2 - 10(2) + 2 &= 12 - 20 + 2 = -6 \\ \text{Tangent: } \frac{y+2}{x-2} = -6 \implies y+2 &= -6x+12 \implies y = 10-6x\\ \text{Normal: } \frac{y+2}{x-2} = \frac{-1}{-6} \implies y+2 &= \frac{x-2}{6} \implies y = \frac{x}{6} - \frac{7}{3} \end{align}$$

Increasing & Decreasing Functions
An increasing function is a function whose gradient is always greater than or equal to zero.

A decreasing function is a function whose gradient is always less than or equal to zero.

Rates of Change
The rate of change of a quantity is the derivative of a quantity with respect to time. For instance, if water were flowing into a bucket at a rate of 1 litre per second, the rate of change of the volume of water in the bucket would be 1 litre per second.

In some cases, the rates of change of two quantities are connected. For example, a circle with a changing radius will have a changing area that depends on the radius. The area of a circle is related to the radius by $$A = \pi r^2$$. If the radius is increasing at a rate of 3 cm/s, its rate of change $$\dfrac{dr}{dt} = 3$$. The rate of change of area can be found using the chain rule:

$$\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt} = \dfrac{d}{dr}(\pi r^2) \times 3 = 2 \pi r \times 3 = 6 \pi r$$

If the radius is increasing at a rate of 3 cm/s, and is currently 5cm, the rate of change of area is $$6 \pi (5) = 30 \pi$$ cm2/s.

Stationary Points


A stationary point is a point on the curve at which the gradient is zero. This means that the derivative of the function at that point is equal to zero.

Stationary points are either maxima or minima. Maxima are where the function reaches a maximum value. Minima are where the function reaches a minimum value. We can determine whether a stationary point is a maximum or minimum by looking at the second derivative.

If the second derivative is positive, the gradient is increasing with input, and thus the stationary point is a minimum.

If the second derivative is negative, the gradient is decreasing with input, and thus the stationary point is a maximum.

Curve Sketching
Stationary points are useful when sketching graphs of curves. By marking the stationary points, we can draw graphs more accurately than otherwise.