Business Mathematics and Statistics · Ch 6 — Differentiation

Derivatives of Exponential and Logarithmic Functions

Beyond powers of $x$ , business and growth problems in this syllabus regularly use the exponential function and the natural logarithm. Their derivatives, together with the power rule already met, complete the standard toolkit needed for this chapter.

Function $f(x)$	Derivative $f'(x)$
$c$ (constant)	$0$
$x^n$	$nx^{n-1}$
$e^x$	$e^x$
$a^x$ ( $a>0$ )	$a^x \ln a$
$\ln x$ (i.e. $\log_e x$ , $x>0$ )	$\dfrac{1}{x}$
$\log_a x$ ( $a>0,\ a\neq1$ )	$\dfrac{1}{x\ln a}$

A distinguishing property of $e^x$ : it is its own derivative — differentiating $e^x$ any number of times still gives $e^x$ back unchanged. This is exactly why $e^x$ appears throughout continuous compound-growth and business-forecasting models.

Combining with the chain rule: when the exponent (for $e^{g(x)}$ ) or the argument of a logarithm (for $\ln(g(x))$ ) is itself a function of $x$ rather than plain $x$ , the chain rule must be layered on top of the standard-function derivative:

\frac{d}{dx}\big(e^{g(x)}\big) = e^{g(x)}\cdot g'(x), \qquad \frac{d}{dx}\big(\ln(g(x))\big) = \frac{g'(x)}{g(x)}

Example: $\dfrac{d}{dx}(e^{4x}) = e^{4x}\times4 = 4e^{4x}$ (here $g(x)=4x$ , $g'(x)=4$ ), and $\dfrac{d}{dx}\big(\ln(7x)\big)=\dfrac{7}{7x}=\dfrac{1}{x}$ (here $g(x)=7x$ , $g'(x)=7$ ). …

Definition 1Derivative of $e^x$

$\dfrac{d}{dx}(e^x)=e^x$ — the exponential function is its own …

Definition 2Derivative of $\ln x$

$\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}$ , fo …

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