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What is the Derivative of ln x/x?

Answer :

The derivative of the function (ln x)/x is (ln x - 1) / x^2.

The derivative of the function (ln x)/x is a common topic in Calculus, and it can be found using the quotient rule.

The quotient rule states that the derivative of the function (f(x))/(g(x)) is given by:

((f(x)g'(x) - f'(x)g(x)) / g(x)^2

Here, f(x) = ln x and g(x) = x.

So, we can use this rule to find the derivative of the function (ln x)/x:

f'(x) = (1/x)

g'(x) = 1

Plugging these values into the quotient rule, we get:

((ln x * 1 - 1/x * x) / x^2)

=((ln x - 1) / x^2)

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Rewritten by : Barada

Final answer:

The derivative of ln x/x is [tex](1 - ln x)/(x^2).[/tex]

Explanation:

The derivative of ln x/x can be found using the quotient rule of differentiation. Let's denote the function as f(x) = ln x/x. Applying the quotient rule, we have:

[tex]f'(x) = (x(d/dx)(ln x) - (ln x)(d/dx)(x))/(x^2)[/tex]

Now, let's evaluate each part separately:

(d/dx)(ln x) = 1/x, as the derivative of natural logarithm is 1/x.

(d/dx)(x) = 1, as the derivative of x with respect to x is 1.

Substituting these values back into the formula, we get:

[tex]f'(x) = (x * 1/x - ln x * 1)/(x^2)[/tex]

Simplifying further, we have:

[tex]f'(x) = (1 - ln x)/(x^2)[/tex]

Therefore, the derivative of ln x/x is [tex](1 - ln x)/(x^2).[/tex]