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Numerade Educator

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Problem 23 Easy Difficulty

If $ f(x) = 5^x $, show that

$ \dfrac{f (x + h) - f(x)}{h} = 5^x \left(\frac{5^h - 1}{h}\right) $

Answer

$5^{x}\left(\frac{5^{h}-1}{h}\right)$

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Video Transcript

Okay, so we have this expression that we want to evaluate for F of X equals five to the X power. So let's go ahead and substitute five to the X in for f of X and five to the X plus H in four f of X plus h sweet five to the X plus H minus five to the X over H. Now we want to simplify that. So remember that when you see five to the X plus H power, it's equivalent to five to the X power power times five to the age power. Think about your exponents rules. And if you were given five to the X times five to the age, you would add the exponents. And now this is still all over H. Okay, so notice that what we can do is factor five to the X power out of both terms in the numerator. And now we have five to the X power multiplied by pi to the H power minus one over H. And that's exactly what we're asked to show