Question

Найти функции f(x)=lnx

Answer 1

Найти производную функцииf(x)=ln(x)
Решение
f′(x)=(ln(x))′=1/x
f′(x)=1/x

Answer 2

Function f(x) = ln(x)

The function f(x) = ln(x) represents the natural logarithm of x. Let's explore some properties and derivatives of this function.

Derivative of f(x) = ln(x)

To find the derivative of f(x) = ln(x), we can use the chain rule. The derivative of ln(x) is given by:

f'(x) = (1/x) * (x)' = 1/x

Therefore, the derivative of f(x) = ln(x) is f'(x) = 1/x.

Other Properties of f(x) = ln(x)

- The domain of the function f(x) = ln(x) is (0, +∞), as the natural logarithm is undefined for x ≤ 0. - The range of the function f(x) = ln(x) is (-∞, +∞), as the natural logarithm takes on all real values. - The function f(x) = ln(x) is strictly increasing, meaning that as x increases, the value of ln(x) also increases. - The graph of f(x) = ln(x) is a curve that starts from negative infinity at x = 0 and approaches positive infinity as x approaches infinity.

Example of Using the Derivative

Let's find the derivative of another function that involves ln(x). Consider the function f(x) = (ln(x))/(3).

To find the derivative of f(x), we can use the quotient rule. The derivative is given by:

f'(x) = [(1/3) * (ln(x))'] - [(ln(x)) * (1/3x)']

Using the derivative of ln(x) from earlier, we have:

f'(x) = (1/3) * (1/x) - (ln(x)) * (1/3x^2)

Simplifying further, we get:

f'(x) = (1/3x) - (ln(x))/(3x^2)

Therefore, the derivative of f(x) = (ln(x))/(3) is f'(x) = (1/3x) - (ln(x))/(3x^2).

I hope this explanation helps! Let me know if you have any further questions.