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Thread: Whats an easy way to remember d/dx of b^x and logb^x

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    Whats an easy way to remember d/dx of b^x and logb^x

    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.
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    Re: Whats an easy way to remember d/dx of b^x and logb^x

    Quote Originally Posted by delgeezee View Post
    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 ...

    $\displaystyle y = b^x$

    $\displaystyle \ln{y} = \ln{b^x}$

    $\displaystyle \ln{y} = x\ln{b}$

    $\displaystyle \frac{d}{dx} \left(\ln{y} = x\ln{b}\right)$

    $\displaystyle \frac{y'}{y} = \ln{b}$

    $\displaystyle y' = y \cdot \ln{b}$

    $\displaystyle y' = b^x \cdot \ln{b}$

    ------------------------------------------------

    $\displaystyle y = \log_b{x}$

    $\displaystyle x = b^y$

    $\displaystyle \frac{d}{dx}\left(x = b^y\right)$

    $\displaystyle 1 = b^y \cdot \ln{b} \cdot y'$

    $\displaystyle 1 = x \cdot \ln{b} \cdot y'$

    $\displaystyle y' = \frac{1}{x \cdot \ln{b}}$
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