Derivatives of Exponentials and Logarithms

Calculus 1 · Axiom Academy

LESSON Derivatives of Exponentials and Logarithms The four exponential and logarithmic derivative rules — and the single picture, slope made of height, that explains every one of them. Start from the limit definition and factor e^x out of the difference quotient. What survives is a single limit that the base e is defined to make equal to 1 — so the derivative is just e^x again. Any base can be rewritten in base e : since , we have . The chain rule then brings down a single constant factor — — which is precisely how much the slope-equals-height picture gets rescaled. is the inverse of e^x , so we differentiate it implicitly. Exponentiate into e^y=x , differentiate both sides, and the slope-equals-height property of e^y does the rest — leaving 1/x . Change of base turns any logarithm into a constant multiple of : . Differentiating, the constant rides along — so the slope is just 1/x scaled down by . You've derived all four exponential and logarithmic derivative rules — and seen that each is the same idea, slope built from height, dressed up by the chain rule or a change of base. Scroll up to revisit any step.

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