The derivative of natural log (y=lnf(x)). Differentiation of lnx.
In this article you will be taught how to differentiate natural log (ln)
If y = lnx then dy/dx is 1/x.
However, this general result is more useful in calculations:
If y = lnf(x), then dy/dx = f´(x)/f(x).
This means that if you differentiate y = lnf(x) you get a fraction, with f´(x) on the numerator and f(x) on the denominator. Let’s go over a few examples of differentiating lnf(x)
Example 1
Work out dy/dx if y = ln(5x)
First identify f(x)
f(x) = 5x
Now work out f´(x):
f´(x) = 5
So put 5 on the numerator and 5x on the denominator:
dy/dx = 5/5x = 1/x
Example 2
Work out dy/dx if y = ln(3x+10)
First identify f(x)
f(x) = 3x + 10
Now work out f´(x):
f´(x) = 3
So put 3 on the numerator and 3x + 10 on the denominator:
dy/dx = 3/(3x+10)
Example 3
Work out dy/dx if y = ln(cos2x)
First identify f(x)
f(x) = cos2x
Now work out f´(x):
f´(x) = -2sin2x
So put -2sin(2x) on the numerator and cos2x on the denominator:
dy/dx = -2sin(2x)/cos2x
Now you should remember that sine divide by cosine gives tan. So dy/dx can be simplified to:
dy/dx = -2tan(2x)
Example 4
Work out dy/dx if y = ln(7x³ + 3x² -9)
First identify f(x)
f(x) = 7x³ + 3x² -9
Now work out f´(x):
f´(x) = 21x² + 6x
So put 21x² + 6x on the numerator and 7x³ + 3x² -9 on the denominator:
dy/dx = (21x² + 6x)/(7x³ + 3x² -9)
Example 5
Work out the gradient of the tangent at x = 2, to the curve y = ln4x
First identify f(x)
f(x) = 4x
Now work out f´(x):
f´(x) = 4
So put 4 on the numerator and 4x on the denominator:
dy/dx = 4/4x = 1/x
All you need to do next is sub x = 2 into dy/dx to give the gradient of the tangent.
m = ½ or 0.5.