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Proofs of derivatives of ln(x) and e^x | Derivatives of common functions | Khan Academy

Can we define e as a number so that the derivative of e^x is equal to the derivative of e^x? Then can we use that to find the derivative of ln x? Also, can we find the value of e using this method by assuming e^x to be a polynomial in the form a0+a1x+a2x^...

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