# 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.

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