Statistics · Ch 9 — Differentiation

Derivatives of Standard Functions

Beyond powers of $x$ , Gujarat board class 12 commerce statistics problems (particularly business and growth applications) also use exponential and logarithmic functions. Their derivatives, alongside the power rule already met, form the complete list of standard derivatives 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. $\log_e x$ , $x>0$ )	$\dfrac{1}{x}$
$\log_a x$ ( $a>0, a \neq 1$ )	$\dfrac{1}{x \ln a}$

Key point about $e^x$ : the exponential function $e^x$ is unique in that it is its OWN derivative — differentiating it any number of times still gives $e^x$ . This special property is exactly why $e^x$ appears throughout continuous compound-growth models in business and economics.

Combining with the chain rule: when the exponent or the argument of a log is itself a function of $x$ (not just $x$ alone), 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: $\dfrac{d}{dx}(e^{3x}) = e^{3x} \times 3 = 3e^{3x}$ (here $g(x)=3x$ , so $g'(x)=3$ ), and $\dfrac{d}{dx}\big(\ln(5x)\big) = \dfrac{5}{5x} = \dfrac{1}{x}$ (here $g(x)=5x$ , so $g'(x)=5$ ). …

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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