Question

How do you find the derivative of $y = \ln \left( {2x} \right)$?

Answer 1

Here we have the two functions which is natural logarithm and other is the function $2x$ that is contained inside the natural logarithm and hence now we need to apply the chain rule when one function is contained inside other:
$\dfrac{d}{{dx}}f(g(x)) = f'\left( {g(x)} \right).g'\left( x \right)$.

Complete step by step solution:
Here we are given to find the derivative of $y = \ln \left( {2x} \right)$
Here we need to know that here in this function given we have one function contained inside the other function. One is the natural logarithm and the other is the function $2x$ whose natural logarithm is taken.
Whenever we are given one function contained inside the other, we need to apply the chain rule for the differentiation which says that:
$\dfrac{d}{{dx}}f(g(x)) = f'\left( {g(x)} \right).g'\left( x \right)$
So when we compare this $f(g(x))$with the given function $\ln \left( {2x} \right)$ we can say that
$g\left( x \right) = 2x \\\ f\left( x \right) = \ln \left( {2x} \right) \\\$
We know that derivative of $\ln \left( x \right) = \dfrac{1}{x}$
Hence when we need to find the derivative of the given term which is $y = \ln \left( {2x} \right)$
We can write that:
$f'\left( {g\left( x \right)} \right) = \dfrac{1}{{2x}} \\\ g'\left( x \right) = 2 \\\$
Now we can substitute these values in the formula and we will get the simplified form of the derivative of $y = \ln \left( {2x} \right)$ as:

$\dfrac{d}{{dx}}\ln \left( {2x} \right) = \dfrac{1}{{2x}}.2$ $= \dfrac{1}{x}$

Note: When we are doing these types of differentiations where we are given to find the derivative of the multiple functions contained one inside the other we just need to differentiate one term after the other according to the chain rule which we have discussed and we will get the required derivative.