How to Differentiate Natural Log in Calculus
A natural logarithm of a function, written ln(x) and sometimes
log(subscript e)(x), is a logarithm that is equal to the power the natural
number, e, would have to be raised to in order to equal x. Given that a
logarithm is essentially a function, this adds some unique issues when
attempting to differentiate natural log.
Sample problem # 1: Differentiate the function ln(x).
To solve this the formal way, using the definition of a derivative, would
take quite a number of steps. Luckily, those that have gone before us found
a nice, succinct role when finding the derivative of a lone natural log.
The rule states d/dx ln(x) = 1/x. Knowing this makes the example incredibly
Step 1: d/dx ln(x) = 1/x
Sample problem # 2: Differentiate the function ln(sqrt(x))
Note:At a glance, this would simply be 1/sqrt(x), but the reality is a bit
trickier than that.
Step 1:Use the law for algorithms that states ln xn = n ln x to separate
the exponent from the function. As sqrt(x) is the same as x1/2, this puts it in a form we can apply the earlier rule to.
d/dx ln(sqrt(x)) = d/dx ½ ln x
Step 2: Find the derivative of the function. Given that ln x derives to 1/x, and a constant’s derivative is always itself:
d/dx ½ ln x = 1/2x
Note: d/dx xy is equal to d/dx x d/dx y.
Tip:Whenever you come across functions you need to differentiate that include natural logs, you can simply substitute 1/x for the derivative of the
natural log of x. It is important to note that this applies specifically to
the natural log, and not to logarithms of any other base.