What is the derivative of ln

Derivation logarithm

In this chapter we look at how to calculate the derivative of the logarithm.

logarithmDerivation logarithm
\ (f (x) = \ ln (x) \)\ (f '(x) = \ frac {1} {x} \)

Remembering the derivation from the logarithm is actually easy. However, if there is not only one \ (x \) as an argument in the logarithm function, it becomes a little more difficult. Then we are forced to fall back on the chain rule.

The chain rule is assumed to be known in the following examples.

example 1

\ [f (x) = \ ln (2x) \]

  • The following applies to the outer function: \ (g (x) = \ ln (x) \ quad \ rightarrow \ quad g '(x) = \ frac {1} {x} \).
  • The following applies to the inner function: \ (h (x) = 2x \ quad \ rightarrow \ quad h '(x) = 2 \).

Now we start accordingly in the formula for the chain rule

\ (f '(x) = g' (h (x)) \ cdot h '(x) \)

\ [f '(x) = \ frac {1} {2x} \ cdot 2 = \ frac {1} {x} \]

Example 2

\ [f (x) = \ ln (x ^ 2 + x) \]

  • The following applies to the outer function: \ (g (x) = \ ln (x) \ quad \ rightarrow \ quad g '(x) = \ frac {1} {x} \).
  • The following applies to the inner function: \ (h (x) = x ^ 2 + x \ quad \ rightarrow \ quad h '(x) = 2x + 1 \).

Now we start accordingly in the formula for the chain rule

\ (f '(x) = g' (h (x)) \ cdot h '(x) \)

\ [f '(x) = \ frac {1} {x ^ 2 + x} \ cdot \ left (2x + 1 \ right) = \ frac {2x +1} {x ^ 2 + x} \]

The examples have shown the great role the chain rule plays in deriving the logarithm. In the case of complex functions in particular, it is worthwhile to first identify the external function and the internal function and derive them separately from one another. Then you put the intermediate results in the formula to get the correct derivation of the logarithm.

Derive logarithms

Below you will find four tutorial videos that explain in detail how to derive logarithms. All of the derivation rules are dealt with using understandable examples.

Factor rule / chain rule

This math video (5:19 min) shows you the application of the factor rule and the chain rule using a logarithm function.

Sum rule / difference rule

This math video (2:18 min) shows you the application of the sum rule and the difference rule using a logarithm function.

Product rule

This math video (2:35 min) shows you the application of the product rule using a logarithm function.

Quotient rule

This math video (2:52 min) shows you the application of the quotient rule using a logarithm function.

More about derivations

There are a few functions that you should know the derivatives of by heart:

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