...Naperian system where u = log p, we shall have du = — . P Therefore the differential of the Naperian logarithm of '' a function, is equal to the differential of the function divided by the function itself. 42. If u = pq where p and q are both functions of the " same independent variable x, then will du =...

...log. a being = 1. From the preceding investigation we learn, that the differential of a logarithmic function is equal to the differential of the function divided by the function itself. (13.) To differentiate an exponential function. 1. Let y = a' then log. y = x log. a .-. d log. y =...

...d log. « = — _ = — . — . ca• с и The differential coefficient therefore of a logarithmic function is equal to the differential of the function divided by the function itself, multiplied by the constant — , the modulus of the system whose base is a. The modulus of the Naperian...

...have seen that when a = 10, - =0 .434294484 с Hence, the differential of a logarithm to the base 10, is equal to the differential of the function divided by the function, multiplied by .434294484 ; consequently, we can change the differential of a logarithm to the base...

...Hence also d. log<f(x) = , — . Hence the differential of the Napierian logarithm of any function of x is equal to the differential of the function divided by the function itself. 36.] The general result of Art. 34 might also have been found from that of Art. 32, as follows : y...

...the logarithm of the expression, for this will separate the factors, and then differentiate. Since the differential of the logarithm of a function is...differential of the function divided by the function, the differentials of the logarithms of the ERRORS. 13 factors will equal their proportional errors,...