Derivative of log base b of x
1/ x ln(b)
The derivative of log base b of x, where b is a positive constant and x is a positive real number, can be found using logarithmic differentiation.
Let y = log_b x. Taking the natural logarithm of both sides, we have:
ln(y) = ln(log_b x)
Using the chain rule and the fact that the derivative of ln(x) is 1/x, we can take the derivative of both sides with respect to x:
1/y * dy/dx = 1/(x *ln(b))
Solving for dy/dx, we get:
dy/dx = (1/y) * (1/(x*ln(b)))
Using the logarithmic identity that y = log_b x, we can substitute back to get the final derivative:
dy/dx = (1/(x * ln(b) * log_b x))
Therefore, the derivative of log base b of x is (1/(x * ln(b) * log_b x)).
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