Derivative of loga x

How to find the derivative of the given function by using the derivative of loga x: formulas, 2 examples, and their solutions.


Derivative of loga x

The derivative of loga x, [loga x]',
is 1/(x ln a).

Derivative of loga |x|

The derivative of loga |x|, [loga |x|]',
is also 1/(x ln a).

Example 1



y = log2 (x3 - 8x) is a composite function of
y = log2 (whole) and (whole) = 2x + 7.

So the derivative of the composite function is,
the derivative of the outer function log2 (x3 - 8x), 1/[(x3 - 8x) ln 2]
the derivative of the inner function x3 - 8x, 3x2 - 8.

Arrange the expression.

Then (3x2 - 8)/[(x3 - 8x) ln 2] is the derivative of log2 (x3 - 8x).

Example 2



Split 5 and x6.
Then log5 5x6 = log5 5 + log5 x6.

Logarithm of a Product

log5 5 = 1

log5 x6 = 6 log5 |x|
The number in the log cannot be minus.
So write the absolute value sign.

Logarithm of a Power

y = 1 + 6 log5 |x|

Then y' = 6⋅[1/(x ln 5)].

The derivative of 1 is 0.
So it's not in y'.

Arrange the right side.

So 6/(x ln 5) is the derivative of log5 5x6.