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.
FormulaDerivative of loga x
The derivative of loga x, [loga x]',
is 1/(x ln a).
Exampley = log2 (x3 - 8x), y' = ?
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.
(3x2 - 8)/[(x3 - 8x) ln 2]
is the derivative of log2 (x3 - 8x).
FormulaDerivative of loga |x|
The derivative of loga |x|, [loga |x|]',
is also 1/(x ln a).
Exampley = log5 5x6, y' = ?
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.
6/(x ln 5)
is the derivative of log5 5x6.