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# Calculus

## Logarithmic Derivative

Using the definition of the derivative, we can show what the derivative of $$y=\log_bx$$ is

\begin{align} y'&=\lim_{h \to 0} \frac{\log_b(x+h)-\log_bx}{h} \\ &=\lim_{h \to 0}\frac{\log_b\frac{x+h}{h}}{h} \end{align}

We multiply $$x$$ into the top and bottom to create a special limit

\begin{align} y'&=\lim_{h \to 0}\frac{1}{x}\frac{x}{h}\log_b\left(1+\frac{x}{h}\right) \\ &=\frac{1}{x}\lim_{h \to 0}\log_b\left(1+\frac{x}{h}\right)^{\frac{x}{h}} \\ &= \frac{1}{x}\log_b\left(\lim_{h \to 0}\left(1+\frac{x}{h}\right)^{\frac{x}{h}}\right) \end{align}

We can show by numerical approximation that

$\lim_{h \to 0}\left(1+\frac{x}{h}\right)^{\frac{x}{h}}=e$

Thus,

$\frac{d}{dx}\log_bx=\frac{1}{x}\log_b e$

For the natural logarithm $$\ln x=\log_e x$$

$\frac{d}{dx}\ln x = \frac{d}{dx}\log_e x=\frac{1}{x}\log_e e=\frac{1}{x}$

## Examples

Example 1: Find the derivative of $y=\log_2 2x^3$

Solution:

\begin{align} y'&= \frac{1}{2x^3}\log_2 e\left(6x^2 \right) \\&=\frac{3}{x}\log_2 e\end{align}

Example 2: Find the derivative of $r=\ln\frac{v^2}{v+2}$

Solution:

\begin{align} r'&=\frac{1}{\frac{v^2}{v+2}}\left(\frac{2v(v+2)-2v}{(v+2)^2} \right) \\&=\frac{v+2}{v^2}\left(\frac{2v(v+1)}{(v+2)^2} \right) \\&=\frac{2(v+1)}{v(v+2)} \end{align}

Example 3: When air friction is considered, the time $$t$$ (in $$s$$) it takes a certain falling object to attain a velocity $$v$$ (in m/s) is given by $$t=5\ln \frac{5}{5-0.1v}$$. Find $$dt/dv$$ for $$v=10.0m/s$$.

Solution: Let's find the derivative of the equation first, then substitute the velocity to find the change at the given moment.

\begin{align} \frac{dt}{dv} &=5\left(\frac{5-0.1v}{5} \right)\left(-0.5(5-0.1v)^{-2} \right) \\ &=-\frac{0.5}{5-0.1v} \end{align}

Now we substitute in $$v=10.0m/s$$

$\frac{dt}{dv}=-\frac{0.5}{5-0.1(10)}=0.125$

The change in time is 0.125 s when the velocity is at 10.0 m/s.