Some Applications of the Chain Rule

$\mathbf{D}(\ln (x))=\dfrac{1}{x} \quad$ and $\quad \mathbf{D}(\ln (g(x)))=\dfrac{\mathbf{g}^{\prime}(x)}{g(x)}$

Proof: We know that the natural logarithm $\ln (x)$ is the logarithm with base $\mathrm{e}$, and $e^{\ln (x)}=x$ for $x > 0$

We also know that $\mathbf{D}\left(\mathrm{e}^{\mathrm{X}}\right)=\mathrm{e}^{\mathrm{x}}$, so using the Chain Rule we have $\mathrm{D}\left(\mathrm{e}^{\mathrm{f}(\mathrm{x})}\right)=\mathrm{e}^{\mathrm{f}(\mathrm{x})} \mathbf{f}^{\prime}(\mathbf{x}) .$ Differentiating each side of the equation $\mathrm{e}^{\ln (\mathrm{x})}=\mathrm{x}$, we get that

The function $\ln (\mathrm{g}(\mathrm{x}))$ is the composition of $\mathrm{f}(\mathrm{x})=\ln (\mathrm{x})$ with $\mathrm{g}(\mathrm{x})$, so by the Chain Rule,

$D \big((\ln (g(x))=D(f(g(x)))=f^{\prime}(g(x)) \cdot g^{\prime}(x)=\frac{1}{g(x)} \cdot g^{\prime}(x)=\frac{g^{\prime}(x)}{g(x)}\big).$

Example 1: Find $\mathbf{D}(\ln (\sin (x)))$ and $\mathbf{D}\left(\ln \left(x^{2}+3\right)\right)$.

Solution: (a) Using the pattern $\mathbf{D}\big((\ln (\mathrm{g}(\mathrm{x}))=\frac{\boldsymbol{g}^{\prime}(\mathrm{x})}{\mathrm{g}(\mathrm{x})} \big)$ with $\mathrm{g}(\mathrm{x})=\sin (\mathrm{x})$, then

$\mathbf{D}(\ln (\sin (x)))=\frac{g^{\prime}(x)}{g(x)}=\frac{D(\sin (x))}{\sin (x)}=\frac{\cos (x)}{\sin (x)}=\cot (x)$

(b) Using the pattern with $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+3$, we have $\mathrm{D}\left(\ln \left(\mathrm{x}^{2}+3\right)\right)=\frac{\mathrm{g}^{\prime}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\frac{2 \mathrm{x}}{\mathrm{x}^{2}+3}$.

We can use the Change of Base Formula from algebra to rewrite any logarithm as a natural logarithm, and then we can differentiate the resulting natural logarithm.

Change of Base Formula for logarithms: $\log _{\mathrm{a}} \mathrm{x}=\frac{\log _{\mathrm{b}} \mathrm{x}}{\log _{\mathrm{b}} \mathrm{a}} \quad$ for all positive $\mathrm{a}, \mathrm{b}$ and $\mathrm{x}$.

Example 2: Use the Change of Base formula and your calculator to find $\log _{\pi} 7$ and $\log _{2} 8$.

Solution: $\log _{\pi} 7=\frac{\ln 7}{\ln \pi} \approx \frac{1.946}{1.145} \approx 1.700 .$ (Check that $\left.\pi^{1.7} \approx 7\right) \log _{2} 8=\frac{\ln 8}{\ln 2}=3$.

Practice 1: Find the values of $\log _{9} 20, \log _{3} 20$ and $\log _{\pi}$ $\mathrm{e}$.

Putting $b=e$ in the Change of Base Formula, $\log _{a} x=\frac{\log _{e} x}{\log _{e} a}=\frac{\ln x}{\ln a}$, so any logarithm can be written as a natural logarithm divided by a constant. Then any logarithm is easy to differentiate.

$\mathbf{D}\left(\log _{a}(x)\right)=\frac{1}{x \ln (a)} \quad$ and $D\left(\log _{a}(f(x))\right)=\frac{f^{\prime}(x)}{f(x)} \cdot \frac{1}{\ln (a)}$

The second differentiation formula follows from the Chain Rule.

Practice 2: Calculate $\mathbf{D}\left(\log _{10}(\sin (x))\right)$ and $\mathbf{D}\left(\log_{\pi}\left(\mathrm{e}^{\mathrm{x}}\right)\right)$.

The number $\mathrm{e}$ might seem like an "unnatural" base for a natural logarithm, but of all the logarithms to different bases, the logarithm with base e has the nicest and easiest derivative. The natural logarithm is even related to the distribution of prime numbers. In 1896, the mathematicians Hadamard and Valle-Poussin proved the following conjecture of Gauss: (The Prime Number Theorem) For large values of $\mathrm{x}, \{$ number of primes less than $\mathrm{x}\} \approx \frac{\mathrm{X}}{\ln (\mathrm{x})}$.