Calculus Small Changes, Big Ideas

Section 4.4 Logarithmic functions

• Differentiating \(\ln x\text{,}\) the inverse of the exponential function:

  \begin{align*} y \amp = \ln x \\ \ee^y \amp = x \\ \dd(\ee^y) \amp = \dd{x} \\ \ee^y \dd{y} \amp = \dd{x} \\ \dd{y} \amp = \frac{1}{\ee^y} \dd{x} \\ \dv{y}{x} \amp = \frac{1}{x} \end{align*}

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• In general, derivatives of inverse functions:

  \begin{align*} y \amp = f\inv(x)\\ f(y) \amp = x\\ \dd(f(y)) \amp = \dd{x}\\ f'(y)\dd{y} \amp = \dd{x}\\ \dd{y} \amp = \frac{1}{f'(y)}\dd{x}\\ \dv{x} f\inv(x) \amp = \frac{1}{f'(f\inv(x))} \end{align*}

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• Logarithmic differentiation:

  \begin{equation*} \dd(\ln y) = \frac{\dd{y}}{y} \end{equation*}

  Good for complicated derivatives or derivatives involving exponents.

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• Reminder of logarithm rules as opposites of exponent rules:

  \begin{align*} \ln(uv) \amp = \ln u + \ln v\\ \ln\left(\frac{u}{v}\right) \amp = \ln u - \ln v\\ \ln(u^r) \amp = r \ln u \end{align*}

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• Differentiating \(x^r\) for any real exponent \(r\text{:}\)

  \begin{align*} y \amp = x^r \\ \ln y \amp = r \ln x \\ \dd(\ln y) \amp = \dd(r \ln x) \\ \frac{1}{y}\dd{y} \amp = r \frac{1}{x} \dd{x} \\ \dd{y} \amp = r x^{r-1} \dd{x} \\ \dv{y}{x} \amp = r x^{r-1} \end{align*}

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• Differentiating \(b^x\) for any base \(b>0\text{:}\)

  \begin{align*} y \amp = b^x \\ \ln y \amp = x \ln b \\ \dd(\ln y) \amp = \dd(x \ln b) \\ \frac{1}{y}\dd{y} \amp = \ln b \dd{x} \\ \dd{y} \amp = y \ln b \dd{x} \\ \dv{y}{x} \amp = b^x \ln b \end{align*}

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• Differentiating \(\log_b{x}\) for any base \(b>0\text{:}\)

  \begin{align*} y \amp = \log_b{x} \\ b^y \amp = x \\ \dd(b^y) \amp = \dd{x} \\ b^y \ln b \dd{y} \amp = \dd{x} \\ \dd{y} \amp = \frac{1}{x \ln b} \dd{x} \\ \dv{y}{x} \amp = \frac{1}{x \ln b} \end{align*}

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• New way of looking at elasticity:

  \begin{equation*} \DS\rdv{y}{x}=\dv{(\ln y)}{(\ln x)}\text{.} \end{equation*}

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  This for example shows that the elasticity of \(uv\) is the sum of the elasticities of \(u\) and \(v\text{.}\)

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