Derivative of Logarithmic Function
f'(x) = 1/x
The logarithmic function is a widely used mathematical function in many fields, including mathematics, physics, engineering, and finance, which is defined as the inverse of the exponential function. It takes a positive real number as an input and returns the power to which a fixed number, known as the base of the logarithm, must be raised to produce the input number.
The derivative of the logarithmic function is calculated using the chain rule of differentiation. Suppose we have a logarithmic function,
f(x) = log_a x
where a is the base of the logarithm. The derivative of the logarithmic function can be written as:
f'(x) = 1/(xln(a))
This means that the derivative of the logarithmic function is equal to the reciprocal of the natural logarithm of the base multiplied by the input value.
We can derive this formula using the chain rule of differentiation, which says that if y = log_a x, then:
dy/dx = (dy/du) * (du/dx)
where u = log_a x. Thus,
dy/du = 1/u(ln(a))
and
du/dx = 1/x(ln(a))
Substituting these values in the chain rule equation, we get:
dy/dx = (1/u(ln(a))) * (1/x(ln(a))) which simplifies to:
dy/dx = 1/(xln(a))
Therefore, the derivative of a logarithmic function is given by 1/(xln(a)).
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