Derivative of logarithmic functions

[Tom McCurdy]

My friend asked for some help with derviatives, I said I would explain here then link him

Here is how you do it

$$\frac {d}{dx} log_b(x) = \frac {1}{xlnb}$$

 

[Tom McCurdy]

so you have at first
$$\frac {d}{dx} log_{10}(10/x) = \frac {1}{\frac{10}{x}ln10}$$

which simplifies to

$$\frac {x}{10ln(10}$$

now you may think your done, but you need to remember the chain rule so you have

$$\frac {x}{10ln(10} + \frac {d}{dx} \frac {10}{x}$$

so let's take the quotient rule and solve for $$\frac {10}{x}$$

f(x) = 10 f'(x)=0
g(x) = x g'(x)=1

g(x)f'(x)-f(x)g'(x)
----------------
g(x)^2

x*0-10*1
---------
x^2

=
$$\frac {-10}{x^2}$$

so then muliply

$$\frac {x}{10ln(10)} * \frac {-10}{x^2}$$

and you will get

$$\frac {d}{dx} log_{10}(10/x)= \frac {-1}{ln(10)*x}$$

 

[Tom McCurdy]

Any other help you could probably use this website address
http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/tutorials/unit3_3.html