Exploring exponential and logarithmic functions as well as L\'Hospitals Rule. He

Question

Exploring exponential and logarithmic functions as well as L'Hospitals Rule. Here are some questions thatwe will address.

Please use examples to illustrate your answers.

Explanation / Answer

(d-dx) ln(x) = lim(d->0) [ ln(x+d) - ln(x) ] / d = lim ln((x+d)/x) / d = lim (1/d) ln(1 + d/x) = lim [ ln (1 + d/x)^(1/d) ]. Set u=d/x and substitute: lim(u->0) [ ln (1 + u)^(1/(ux)) ] = 1/x ln [ lim(u->0) (1 + u)^(1/u) ] = 1/x ln (e) (Definition of e) = 1/x. and (1) (d-dx) ln(e^x) = (d-dx) x = 1 (d-dx) ln(e^x) = (d/du) ln(u) (d-dx) e^x (Set u=e^x) = 1/u (d-dx) e^x = 1/e^x (d-dx) e^x = 1 (equation 1) (d-dx) e^x = e^x Q.E.D.

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