Subsection 6.2.1 Derivatives of Logrithmic Functions
If \(f(x) = \ln(x)\text{,}\) then what is \(f'(x)\text{?}\)
\begin{align*}
f'(x) \amp = \lim\limits_{h\to 0} \dfrac{f(x+h) - f(x)}{h} \\
\amp = \lim\limits_{h\to 0} \dfrac{\ln(x+h) - \ln(x)}{h} \\
\amp = \lim\limits_{h\to 0} \dfrac{\ln(\dfrac{x+h}{x})}{h} \\
\amp = \lim\limits_{h\to 0} \dfrac{\ln(\dfrac{x}{x}+\frac{h}{x})}{h} \\
\amp = \lim\limits_{h\to 0} \dfrac{\ln\left(1+\dfrac{h}{x}\right)}{h} \\
\amp = \lim\limits_{h\to 0} \dfrac{1}{h} \ln\left(1 + \dfrac{h}{x}\right) \\
\amp = \lim\limits_{v\to 0} \dfrac{1}{vx} \ln(1+v) \\
\amp = \lim\limits_{v\to 0} \dfrac{1}{v} \cdot \dfrac{1}{x} \ln(1+v) \\
\amp = \dfrac{1}{x} \lim\limits_{v\to 0} \dfrac{1}{v} \ln(1+v) \\
\amp = \dfrac{1}{x} \lim\limits_{v\to 0} \ln\left((1+v)^{\dfrac{1}{v}}\right)\\
\amp = \dfrac{1}{x} \ln\left(\lim\limits_{v\to 0} (1+v)^{\dfrac{1}{v}}\right) \\
\amp = \dfrac{1}{x} \ln(e) \\
\amp = \dfrac{1}{x} (1) \\
f'(x)\amp = \dfrac{1}{x}
\end{align*}
Definition 6.2.1. General Definition of the Derivative of ln(x).
A more general form of \(\dfrac{d}{dx}(\ln(x)) = \dfrac{1}{x}\) is \(\dfrac{d}{dx}(\ln(u)) = \dfrac{1}{u} \cdot \dfrac{du}{dx}\text{.}\)
Since \(\dfrac{d}{dx}(\ln(x)) = \dfrac{1}{x}\text{,}\) \(\displaystyle \int \dfrac{1}{x} dx = \ln(x) + c\text{.}\) Power Rule for Integrals.
Recall, the power rule for integrals:
\begin{equation*}
\int x^n dx = \dfrac{x^{n+1}}{n+1} + c
\end{equation*}
* The only exception to the power rule is when \(n = -1\text{.}\)
\begin{equation*}
\int \dfrac{1}{x} dx = \int x^{-1} dx \neq \dfrac{x^0}{0} + c
\end{equation*}
Subsection 6.2.2 Logrithmic Differentiation
How would we find \(\dfrac{dy}{dx}\) if \(y = \dfrac{(x^4 + 7)\sqrt{x}}{\sin(x)}\text{?}\)
This can be done in three steps:
take the natural log of both sides,
take the derivative of both sides with respect to x,
multiply both sides times y.
Example 6.2.1.
Using the equation above:
\begin{align*}
y \amp = \dfrac{(x^4 + 7) \sqrt{x}}{\sin(x)}\\
\ln(y) \amp = \ln \left( \dfrac{(x^4 + 7) \sqrt{x}}{ \sin(x)} \right) \\
\amp = \ln((x^4 + 7)\sqrt{x})-\ln(\sin(x)) \\
\amp = \ln(x^4 + 7) + \ln(\sqrt{x})-\ln(\sin(x)) \\
\amp = \ln(x^4 + 7) + \frac{1}{2}\ln(x)-\ln(\sin(x))\\
\frac{1}{y} \cdot \dfrac{dy}{dx} \amp = \dfrac{1}{x^4+7} \cdot 4x^3 + \frac{1}{2} \cdot \frac{1}{x} - \dfrac{1}{\sin(x)} \cdot \cos(x) \\
\dfrac{dy}{dx} \amp = \left( \dfrac{1}{x^4+7} \cdot 4x^3 + \frac{1}{2} \cdot \frac{1}{x} - \dfrac{1}{\sin(x)} \cdot \cos(x) \right) \cdot y \\
\dfrac{dy}{dx} \amp = \left( \dfrac{4x^3}{x^4+7} + \frac{1}{2x} - \dfrac{\cos(x)}{\sin(x)} \right) \cdot \left( \dfrac{(x^4+7)(\sqrt{x})}{\sin(x)} \right)
\end{align*}