The derivative of with respect to is
We use the logarithm to compute the derivative of a function.
Logarithmic Differentiation
In this section we use the natural logarithm and its properties to help us to compute the derivative of some special functions as well as some complicated functions. From the chain rule, we have If we solve this equation for we get the logarithmic differentiation formula: It says that we can find the derivative of a function by multiplying the original function by the derivative of its logarithm. The reason that this formula is potentially useful is that the logarithm function has three calculus friendly properties:
\begin{align*} 1.\;\; &\ln (uv) = \ln (u) + \ln (v)\\ 2.\;\; &\ln \left (\frac{u}{v}\right ) = \ln (u) - \ln (v)\\ 3.\;\; &\ln (u^n) = n\ln (u). \end{align*}
In these pre-calculus formulas, the expressions and can be functions as well as constants.
Note that we used the fact that the logarithm and the exponential functions are inverses in the last line to write . Now we can use the chain rule to compute the derivative of . We have \begin{align*} \frac{d}{dx}\left [ \ln (g(x))\right ] &= \frac{d}{dx}\left [ x + 4\ln (\sin (x)) - \frac 12 \cdot \ln (x^4 + x^2 + 1)\right ]\\ &= 1+ 4 \cdot \frac{\cos (x)}{\sin (x)} - \frac 12 \cdot \frac{4x^3 + 2x}{x^4 + x^2 + 1}. \end{align*}
Finally, we use the logarithmic differentiation formula to obtain . With a couple of minor simplifications, we have:
As a final note, even though is only defined for , the logarithmic differentiation formula given above will yield the correct result even in situations where . This is because and so, by the chain rule,
2024-09-27 13:55:09