Question

Помогите найти производную функции: f(x)=sinx/x^2

Answer 1

F'(x)=x^2*cosx-2x*sinx/x^4

Answer 2

F`(x)=[(sinx)`*x²-(x²)`*sinx]/x^4=(x²*cosx-2x*sinx)/x^4=x*(x*cosx-2sinx)/x^4=
=(x*cosx-2sinx)/x³

Answer 3

Derivative of the Function f(x) = sin(x)/x^2

To find the derivative of the function \( f(x) = \frac{sin(x)}{x^2} \), we can use the rules of differentiation. The derivative of a function represents the rate of change of the function with respect to the independent variable.

The derivative of the function \( f(x) = sin(x) \) is \( f'(x) = cos(x) \). The derivative of the function \( g(x) = x^2 \) is \( g'(x) = 2x \).

Using the quotient rule for differentiation, the derivative of the given function can be calculated as follows: \[ f(x) = \frac{u}{v} \] \[ f'(x) = \frac{u'v - uv'}{v^2} \]

Where: - \( u = sin(x) \) and \( u' = cos(x) \) - \( v = x^2 \) and \( v' = 2x \)

Therefore, the derivative of the function \( f(x) = \frac{sin(x)}{x^2} \) is: \[ f'(x) = \frac{cos(x) * x^2 - sin(x) * 2x}{(x^2)^2} \] \[ f'(x) = \frac{x^2 * cos(x) - 2x * sin(x)}{x^4} \]

So, the derivative of the function \( f(x) = \frac{sin(x)}{x^2} \) is \( f'(x) = \frac{x^2 * cos(x) - 2x * sin(x)}{x^4} \).

I hope this helps! If you have further questions, feel free to ask.