7. Logarithmic Differentiation Calculator — Theory
Logarithmic differentiation is a technique that simplifies differentiation by taking the natural logarithm first.
It is especially useful when:
- you have a variable exponent like \(u(x)^{v(x)}\) (both base and exponent depend on \(x\)),
- you have a long product or quotient of many factors,
- you want cleaner algebraic steps (turning products into sums).
1) The core idea
Suppose
\[
y = f(x).
\]
Take logarithms:
\[
\ln|y| = \ln|f(x)|.
\]
We use \(\ln|\,\cdot\,|\) to handle sign issues and make the step valid on intervals where \(y\neq 0\).
At points where \(f(x)=0\), \(\ln|f(x)|\) is undefined, so the log-step is not valid exactly at those zeros.
Differentiate both sides:
\[
\frac{d}{dx}\ln|y| = \frac{d}{dx}\ln|f(x)|.
\]
Using the derivative rule \(\dfrac{d}{dx}\ln|y|=\dfrac{y'}{y}\) (where \(y\neq 0\)), we get:
\[
\boxed{\frac{y'}{y} = \frac{d}{dx}\ln|f(x)|}.
\]
Often we further use \(\dfrac{d}{dx}\ln|f| = \dfrac{f'}{f}\) (where \(f\neq 0\)), giving:
\[
\boxed{\frac{y'}{y}=\frac{f'}{f}}
\quad\Rightarrow\quad
\boxed{y' = y\cdot\frac{f'}{f}}.
\]
2) Variable exponent form \(u(x)^{v(x)}\)
This is the classic case where logarithmic differentiation is much easier than trying to use other rules directly.
Let
\[
y = u(x)^{v(x)}.
\]
Then
\[
\ln|y| = v(x)\,\ln|u(x)|.
\]
Differentiate:
\[
\frac{y'}{y} = v'(x)\ln|u(x)| + v(x)\frac{u'(x)}{u(x)}.
\]
Multiply by \(y=u^v\):
\[
\boxed{
y' = u(x)^{v(x)}\left(v'(x)\ln|u(x)| + v(x)\frac{u'(x)}{u(x)}\right).
}
\]
3) Products and quotients
If
\[
y = \frac{u_1u_2\cdots u_m}{v_1v_2\cdots v_n},
\]
then logarithms turn products into sums:
\[
\ln|y| = \sum_{i=1}^{m}\ln|u_i| - \sum_{j=1}^{n}\ln|v_j|.
\]
Differentiating gives:
\[
\frac{y'}{y} = \sum_{i=1}^{m}\frac{u_i'}{u_i} - \sum_{j=1}^{n}\frac{v_j'}{v_j}.
\]
Finally:
\[
\boxed{
y' = y\left(\sum_{i=1}^{m}\frac{u_i'}{u_i} - \sum_{j=1}^{n}\frac{v_j'}{v_j}\right).
}
\]
4) Worked example: \(y=(\sin x)^x\)
Let \(y=(\sin x)^x\). Take logs:
\[
\ln|y| = x\ln|\sin x|.
\]
Differentiate:
\[
\frac{y'}{y} = \ln|\sin x| + x\cdot\frac{\cos x}{\sin x}.
\]
Multiply by \(y\):
\[
\boxed{
y' = (\sin x)^x\left(\ln|\sin x| + x\cot x\right).
}
\]
5) What the calculator shows
- It computes \(f'(x)\) symbolically.
- It generates a step-by-step log-differentiation outline (variable-exponent and product/quotient special cases).
- It plots \(f\) and \(f'\) side-by-side with shared pan/zoom and auto-fit.
