Basic doubt on differential of ln(ax) from first principles

[Srinivasa B]

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

[ajb]

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.

[Dave]

You'll also find it useful to note that

 

[math]\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x[/math].

[Srinivasa B]

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