3.7: Derivatives of the Logarithm

The function $y = \log_b x$ seems to be differentiable on it's domain. We will show this to be true.

Differentiating Logarithmic Functions

\[\dfrac{d}{dx}\log_b x = \dfrac{1}{x \ln b}\]

In general, $\dfrac{d}{dx} \ln x = \dfrac{1}{x}$.

Differentiate the following:
  1. $y = \ln (x^3 + 1)$
  2. $y = \ln (\sin x)$
  3. $y = \sqrt{\ln x}$
  4. $f(x) = \log_{10}(2 + \sin x)$
  5. $y = \ln\left(\dfrac{x + 1}{\sqrt{x - 2}}\right)$
If $f(x) = \ln \lvert x \rvert$, find $f'(x)$.

Logarithmic Differentiation

We can use logarithms and implicit differentiation to quickly take the derivative of a complicated looking function.

Differentiate the following function \[y = \dfrac{x^{3/4}\sqrt{x^2 + 1}}{(3x + 2)^5}\] by first taking the natural log of both sides, then implicitly differentiating.
Differentiate the following function \[y = (2x + 1)^5(x^4-3)^6\] using logarithmic differentiation.
Differentiate the following function \[y = \sqrt[4]{\dfrac{x^2 + 1}{x^2 - 1}}\] using logarithmic differentiation.
Show the Power Rule \[\dfrac{d}{dx} x^n = nx^{n-1}\] is true with logarithmic differentiation.
Differentiate $y = x^{\sqrt{x}}$.

The Number $e$ as a Limit

We will now define Euler's Number $e$. Previously we said

$e$ is the number where \[\lim_{h\rightarrow 0}\dfrac{e^h - 1}{h} = 1\]

With the derivative of $\ln x$, we can show:

\[e = \lim_{x\rightarrow 0}(1 + x)^{\frac{1}{x}}\]