Srinivasa B Posted July 19, 2010 Share Posted July 19, 2010 Hi Guys, I have a fundamental doubt on differential of y = f(x) = ln(ax) where, ln - natual logorithm to base e, a - constant x - independent variable. I need to find the value of dy/dx from first principles. Can anybody help me on this? Thanks, Srini Link to comment Share on other sites More sharing options...
ajb Posted July 19, 2010 Share Posted July 19, 2010 Write out dy/dx using the definition of the derivative. Then think about what properties of logarithms could be useful in simplifying what you get. What happens when h (or dx) is small? Let us know how far you get. Link to comment Share on other sites More sharing options...
Dave Posted July 19, 2010 Share Posted July 19, 2010 You'll also find it useful to note that [math]\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x[/math]. Link to comment Share on other sites More sharing options...
Srinivasa B Posted July 20, 2010 Author Share Posted July 20, 2010 Ya, if y = lx(x), y = ln(ax) = ln(a) + ln(x) so, dy/dx = 0 + lim (ln(x + dx) - ln(x)) / dx dx -> 0 = lim ln(1+ dx/x) / dx dx -> 0 now, to get into e pow x form, put n = x/dx now, dy/dx = (1/x) * lim ( ln(1 + 1/n) ^ n) n -> inf so, dy/dx = 1/x. Yuppie Link to comment Share on other sites More sharing options...
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