Business Mathematics and Statistics · Ch 5 — Differential Calculus (Functions & Graphs, Limits & Derivatives, Differentiation Techniques)

Rules of Differentiation — Sum, Difference, Product and Quotient

Real functions in business problems are built by combining simpler functions — added, subtracted, multiplied, or divided — so the next step is a set of rules for differentiating such combinations directly, without returning to first principles every time. Let $u=u(x)$ and $v=v(x)$ be two differentiable functions of $x$ .

Sum and difference rule:

\frac{d}{dx}\big(u \pm v\big) = \frac{du}{dx} \pm \frac{dv}{dx}

Differentiation 'distributes' over addition and subtraction — differentiate each term separately and combine.

Product rule. The derivative of a product is not simply the product of the derivatives:

\frac{d}{dx}(uv) = u\,\frac{dv}{dx} + v\,\frac{du}{dx}

In words: (first function) × (derivative of second) + (second function) × (derivative of first).

Quotient rule. For $v \neq 0$ :

\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^2}

A useful way to remember the order in the numerator: it is (denominator)×(derivative of numerator) minus (numerator)×(derivative of denominator), all over (denominator) $^2$ — the minus sign, and the order, both matter.

Alongside the power rule from the previous section, the following standard derivatives are used freely from here on:

$f(x)$	$f'(x)$
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$\dfrac1x$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$

Definition 1Product Rule

For differentiable $u(x), v(x)$ : $\dfrac{d}{dx}(uv) = u\,v' + v\,u'$ — (first)×(derivative of second) plus (second)×( …

Definition 2Quotient Rule

For differentiable $u(x), v(x)$ with $v\neq0$ : $\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) = \dfrac{v\,u' - u\,v'}{v^2}$ — (denominator × derivative of numerator, minus numerator × derivative of deno …

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