Whats an easy way to remember $\displaystyle \frac{d}{dx}[b^x] $and $\displaystyle \frac{d}{dx} logb ^x$ besides remembering the d/dx formulas for them?
I am in calc 2 and I forgot! We haven't used them yet, thankfully.
Whats an easy way to remember $\displaystyle \frac{d}{dx}[b^x] $and $\displaystyle \frac{d}{dx} logb ^x$ besides remembering the d/dx formulas for them?
I am in calc 2 and I forgot! We haven't used them yet, thankfully.
derive them yourself ...
$\displaystyle y = b^x$
$\displaystyle \ln{y} = \ln{b^x}$
$\displaystyle \ln{y} = x\ln{b}$
$\displaystyle \frac{d}{dx} \left(\ln{y} = x\ln{b}\right)$
$\displaystyle \frac{y'}{y} = \ln{b}$
$\displaystyle y' = y \cdot \ln{b}$
$\displaystyle y' = b^x \cdot \ln{b}$
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$\displaystyle y = \log_b{x}$
$\displaystyle x = b^y$
$\displaystyle \frac{d}{dx}\left(x = b^y\right)$
$\displaystyle 1 = b^y \cdot \ln{b} \cdot y'$
$\displaystyle 1 = x \cdot \ln{b} \cdot y'$
$\displaystyle y' = \frac{1}{x \cdot \ln{b}}$