im stuck here, how do i do this?
g(x) = ( log [10] x ) / (x^2)
note:
log [10] x-- this is log, sub 10, x... i didnt know how to write it out
Hello,
not quite. Remember $\displaystyle \frac{1}{\ln(10)}$ is a constant.
$\displaystyle g(x) = \frac {1}{\ln 10} \cdot \frac {\ln x}{x^2}$ will give:
$\displaystyle g'(x) = \frac {1}{\ln 10} \cdot \frac {x^2 \cdot \frac{1}{x} - \ln (x) \cdot 2x}{x^4}$
I'll leave the rest for you. (You can cancel out a factor - under certain conditions!)
as earboth already said, when we have a constant multiplier to a derivative, we can essentially forget about it and differentiate the (variable) function and then apply the constant multiplier when we're done. in fact, this is one of the properties of derivatives:
$\displaystyle \frac {d}{dx} c \cdot f(x) = c \frac {d}{dx}f(x)$
where $\displaystyle c$ is a constant
Hello, runner07!
You should have been shown the formula for "other bases".
You can always derive the formula . . . (Yeah, right!)im stuck here, how do i differentiate this?
. . $\displaystyle g(x) \:= \:\frac{\log_{10}x}{x^2}$
We have: .$\displaystyle y \:=\:\log_b(x)$ . ... where $\displaystyle b$ is a positive real number $\displaystyle \neq 1$
. . Re-write in exponential form: .$\displaystyle b^y \:=\:x$
. . Take logs: .$\displaystyle \ln\left(b^y\right) \:=\:\ln(x)\quad\Rightarrow\quad y\!\cdot\!\ln(b) \:=\:\ln(x)$
. . Differentiate implicitly: .$\displaystyle \frac{dy}{dx}\!\cdot\ln(b) \:=\:\frac{1}{x}$
Therefore: . $\displaystyle \boxed{\frac{dy}{dx} \;=\;\frac{1}{\ln b}\!\cdot\!\frac{1}{x}} $
So the derivative of $\displaystyle \log_{10}x$ is: .$\displaystyle \frac{1}{\ln10}\!\cdot\!\frac{1}{x}$
You can use the product rule too
$\displaystyle g(x)=\frac{\log_{10}x}{x^2}=\frac{\dfrac{\ln x}{\ln10}}{x^2}=\frac1{\ln10}\cdot\frac{\ln x}{x^2}$
So, all we have to do it's take the derivative of $\displaystyle \frac{\ln x}{x^2}=x^{-2}\ln x$, now by the product rule
$\displaystyle \ln 10\cdot g'(x)=-2x^{-3}\ln x+x^{-2}\cdot\frac1x=-\frac{2\ln x}{x^3}+\frac1{x^3}=\frac{1-2\ln x}{x^3}$