Question

Почему, находя f(x)=√x, мы получаем производную, равную 1/2√x? Если кто сможет, объясните пожалуйста.

Answer 1

 f⁾(x)=(√x )⁾= (х¹/²)⁾ = (1/2)х¹/² ⁻ ¹ =(1/2)х ⁻¹/²  = 1/(2х¹/²) =1/(2√x)

Answer 2

Understanding the Derivative of f(x) = √x

When we calculate the derivative of the function f(x) = √x, we get a result of 1/(2√x). This can be explained through the process of finding the derivative of a square root function.

The derivative of a function represents the rate of change of the function with respect to its variable. In the case of f(x) = √x, we can use the power rule for differentiation to find its derivative.

The power rule states that if we have a function of the form f(x) = x^n, then its derivative is given by f'(x) = nx^(n-1).

In the case of f(x) = √x, we can rewrite it as f(x) = x^(1/2). Applying the power rule, the derivative of f(x) = √x is:

f'(x) = (1/2)x^(-1/2) = 1/(2√x).

This result shows that the derivative of f(x) = √x is 1/(2√x), which aligns with the standard derivative rule for the square root function.

In summary, the derivative of f(x) = √x is 1/(2√x) because of the application of the power rule for differentiation, which yields the rate of change of the square root function with respect to its variable.

I hope this explanation helps! If you have further questions or need more clarification, feel free to ask.