Dr Phill’s Science Made Simple

Mathematics

Derivatives

The derivative, or differential, of a function of a variable \(y = f(x)\) for a particular value \(x = a\), if it exists, defines the gradient of the tangent to the curve at that point.

Differential calculus was discovered independently by Gottfired Leibnitz and Isaac Newton in the late 17th century. Newton wrongly accused Leibnitz of plagiarism and it was a major scandal. In fact it is Leibnitz' notation that is in common usage today.

In physics, functions are often function of time. The function then corresponds to distance, the first derivative is velocity and the second derivative is acceleration.

Notation

The Liebnitz notation, introduced in 1675, is commonly used.

\[\frac{dy}{dx} \equiv \frac{d}{dx}f(x)\]

Higher orders of deriviation use a number suffix. It is effectively raining the d symbols to a power.

\[\frac{d^2y}{dx^2} \equiv \frac{d^2}{dx^2}f(x)\]

There is also the prime mark or dash notation, introduced by Joseph-Louis Lagrange, \(y' = f'(x)\), \(y'' = f''(x)\).

Newton used the dot notation \(\dot{y}\), \(\ddot{y}\). It is only used for time derivatives and doesn’t lend itself to orders higher than three.

First Principles

Differentiation can be defined from first principles. Given that \(y\) is a function of \(x\).

\[y = f(x)\]

If the value of \(x\) is incremented by a small amount \(\delta x\), then \(y\) will be incremented by a corresponding amount \(\delta y\).

\[y + \delta y = f(x + \delta x)\]

Subtract Equation (3) from Equation (4) and divide by \(\delta x\).

\[\frac{\delta y}{\delta x} = \frac{f(x + \delta x) - f(x)}{\delta x}\]

Now, take the limit as \(\delta x\rightarrow 0\) to get the derivative. The increment \(\delta x\) is often replaced by \(h\).

\[\frac{dy}{dx} = \lim\limits_{\delta x \rightarrow 0}\frac{f(x + \delta x) - f(x)}{\delta x}\]

Example \(y = x^2\).

\[\frac{dy}{dx} = \lim\limits_{\delta x \rightarrow 0}\frac{(x + \delta x)^2 - x^2}{\delta x} = \lim\limits_{\delta x \rightarrow 0}\frac{x^2 + 2x\delta x + \delta x^2 - x^2}{\delta x} = \lim\limits_{\delta x \rightarrow 0}\frac{2x\delta x + \delta x^2}{\delta x} = \lim\limits_{\delta x \rightarrow 0}2x + \delta x = 2x\]

This can be extended to show that if \(y = x^n\).

\[\frac{dy}{dx} = nx^{n-1}\]

Products and Quotients

If \(u\) and \(v\) are functions of \(x\) and \(y = uv\). Apply first principles.

\[\frac{dy}{dx} = \lim\limits_{\delta x \rightarrow 0}\frac{(u + \delta u)(v + \delta v) - uv}{\delta x} = \lim\limits_{\delta x \rightarrow 0}\frac{u\delta v + \delta uv + \delta u \delta v}{\delta x} = u\frac{dv}{dx} + v\frac{du}{dx}\]

If \(u\) and \(v\) are functions of \(x\) and \(y = u/v\). Apply first principles.

\[\frac{dy}{dx} = \lim\limits_{\delta x \rightarrow 0}\frac{1}{\delta x}\left(\frac{u + \delta u}{v + \delta v} - \frac{u}{v}\right) = \lim\limits_{\delta x \rightarrow 0}\frac{1}{\delta x}\left(\frac{v(u + \delta u) - u(v + \delta v)}{v(v + \delta v)}\right) = \lim\limits_{\delta x \rightarrow 0}\frac{1}{\delta x}\left(\frac{v\delta u - u\delta v}{v^2 + v\delta v}\right) = \frac{1}{v^2}\left(v\frac{du}{dx} - u\frac{dv}{dx}\right)\]