I was watching patrickJMT's derivates of logarithmic functions video and I got stuck understanding why these derivates are treated differently:
$f(x) = \ln(g(x))$ derivative is $f'(x) = \frac{1 }{ g(x)} * g'(x)$
while the function $f(x) = log_a g(x)$ has the derivative $f'(x) = \frac{1 }{ g(x) \ln(a)}$.
Why do we not include the derivative of $g(x)$ as well to the case of log function?
Best Answer
You are correct: the derivative of $\log_a(g(x))$ is in fact $\displaystyle{\frac{g'(x)}{g(x)\ln a}}$. This is because, by the chain rule $$ \frac{d}{dx}\left(\log_a(g(x))\right)=(\log_a)'(g(x))\cdot g'(x)=\frac{1}{u\ln a}\bigg\rvert_{u=g(x)}\cdot g'(x)=\frac{g'(x)}{g(x)\ln a} \, . $$ Note that since $f$ and $g$ are functions of one variable, their partial derivatives are exactly the same as their total derivatives. The concept of a "partial derivative" is only useful when considering functions that depend on several variables, such as $f(x,y)=xy+y^3$.