Question

Найти производную степенной функции: y=sqrt(2x)

Answer 1

$( \sqrt{2x} )`= (\sqrt{2}* \sqrt{x} )`= \sqrt{2}* (\sqrt{x} )`= \frac{ \sqrt{2} }{2 \sqrt{x} }$

Answer 2

У меня получилось вот так

Answer 3

Finding the Derivative of the Power Function: y = sqrt(2x)

To find the derivative of the power function y = sqrt(2x), we can use the power rule of differentiation. The power rule states that if we have a function of the form y = x^n, where n is a constant, then the derivative of y with respect to x is given by:

dy/dx = n * x^(n-1)

In this case, we have y = sqrt(2x), which can be written as y = (2x)^(1/2). Applying the power rule, we get:

dy/dx = (1/2) * (2x)^(1/2 - 1)

Simplifying further, we have:

dy/dx = (1/2) * (2x)^(-1/2)

Now, let's simplify the expression (2x)^(-1/2). Using the property (a^m)^n = a^(m*n), we can rewrite it as:

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

Simplifying further, we have:

(2x)^(-1/2) = (1/sqrt(2)) * (1/sqrt(x))

Substituting this back into the derivative expression, we get:

dy/dx = (1/2) * (1/sqrt(2)) * (1/sqrt(x))

Therefore, the derivative of the power function y = sqrt(2x) is:

dy/dx = (1/2) * (1/sqrt(2)) * (1/sqrt(x))

Please note that this is a general expression for the derivative of the power function y = sqrt(2x). If you have a specific value of x, you can substitute it into the expression to find the derivative at that point.