Notes on the derivative

I’m going to keep some notes on the derivative here. This page should grow while we discuss the basics of the derivative.

What is a derivative?

Given a function \(f:[a,b]\to\mathbb{R}\), the derivative of \(f\) is a new function \(f'\) that tells us some important information about the original - in particular, \(f'\) tells us the instantaneous rate of change of its output relative to its input.

Interpretations of the derivative

Slope

If \(y=f(x)\) defines a graph in the plane, the \(f'(x)\) tells us the rate of change of \(y\) with respect to \(x\) right at a single point. We can think of this as slope of a tangent line.

{
    const fpts = d3.range(-1,5.5,0.01).map(x => [x,(x**2 - 3*x)/4])
    const l = x => 1 + 1.25*(x-4);
    const lpts = d3.range(2,6.5,1).map(x => [x,l(x)])

    return Plot.plot({
    y: {domain: [-1.2,3.8]},
    x: {domain: [-1.2,5.8]},
    marks: [
        Plot.line(fpts, {strokeWidth: 3}),
        Plot.line(lpts, {strokeWidth: 2}),
        Plot.dot([[4,1]], {fill: 'gray', r: 5, stroke: 'currentColor'}),
        Plot.ruleX([0]),
        Plot.ruleY([0]),
        Plot.axisX({y:0, ticks: [-1,1,2,3,4,5], tickFormat: d => parseInt(d)}),
        Plot.axisY({x:0, ticks: [-1,1,2,3], tickFormat: d => parseInt(d)}),
    ]
    })
}

Velocity and acceleration

Suppose an object moves along a horizontal axis and it’s position \(x\) is given by a function \(x(t)\) of time \(t\). Then \(x'(t)\) tells us the instantaneous rate of change of the position with respect to time. This is called the velocity.

If we denote the velocity \(v\) as a function of time \(t\) by \(v(t)\), then we can apply the derivative again to get \(v'(t)\). This is called the acceleration.

The definition of the derivative

Given a function \(f\) of the variable \(x\), it’s derivative \(f'(x)\) is defined by \[ f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}. \]

The expression \[ \frac{f(x+h)-f(x)}{h} \] is called the difference quotient of the function \(f\).

An example

Suppose that \(f(x) = 3x^2 - x\). Then, its difference quotient is \[ \frac{(3(x+h)^2 - (x+h)) - (3x^2 - x)}{h}. \] If we fully expand the numerator, we get \[ \frac{3x^2 + 6xh + 3h^2 - x-h - 3x^2 + x}{h}. \] If we cancel terms and factor an \(h\) out of the numerator, we get \[ \frac{6xh + 3h^2 - h}{h} = \frac{(6x + 3h - 1)h}{h}. \] At that point, we can cancel the \(h\) to get an expression whose limit is easy to compute: \[ (6x+3h-1) \to 6x-1 \text{ as } h\to0. \]

We can string that into a whole sequence of equalities like so:

\[ \begin{aligned} f'(x) &= \lim_{h\to0} \frac{(3(x+h)^2 - (x+h)) - (3x^2 - x)}{h} \\ &= \lim_{h\to0} \frac{3(x^2 + 2xh + h^2) - x-h - 3x^2 + x}{h} \\ &= \lim_{h\to0} \frac{3x^2 + 6xh + 3h^2 - x-h - 3x^2 + x}{h} \\ &= \lim_{h\to0} \frac{6xh + 3h^2 - h}{h} = \lim_{h\to0} \frac{(6x + 3h - 1)h}{h} \\ &= \lim_{h\to0} (6x+3h-1) = 6x-1. \end{aligned} \]

Differentiation rules

Differentiation rules are tools to help us compute derivatives efficiently. They come in two main classes:

Basic rules that tell us how to compute the derivative of a single function (like \(\sin(x)\)) or family of functions dependant on a parameter (like \(x^n\)).
Combination rules that tell us how to compute the derivative of a function formed by algebraically combining functions whose derivatives we know.

Basic rules

This list will grow!

The power rule

If \(n\in\mathbb{N}\), then \(\frac{d}{dx} x^n = nx^{n-1}\)

Example: \(\frac{d}{dx}x^{42} = 42x^{41}\)

Combination rules

This list will also grow!

The constant multiple rule

If \(f\) is a differentiable function and \(c\in\mathbb{R}\), then \(\frac{d}{dx} c f(x) = c f'(x)\).

Example: \(\frac{d}{dx}(2x^{42}) = 84x^{41}\)

The constant multiple rule

If \(f\) and \(g\) are differentiable functions, then \(\frac{d}{dx} (f(x)+g(x)) = f'(x) + g'(x)\).

Example: \(\frac{d}{dx}(x^{42} + x^2) = 42x^{41} + 2x\)

More examples

\(\frac{d}{dx}(2x^{42} + 3x^2) = 84x^{41} + 6x\)

Proofs

The differentiation rules can all be proved using the definition of the derivative!!