Find the derivative of y=xsin(2/x)? On web-assign everything that I have tried has been wrong.
Use the product rule
dy/dx = x(cos(2/x)(-2/x^2)) + 1(sin(2/x)
= (-2/x)cos(2/x) + sin(2/x)
To find the derivative of the function y = x*sin(2/x), we can use the product rule and the chain rule.
The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(d/dx)(u(x)*v(x)) = u(x)*(d/dx)(v(x)) + v(x)*(d/dx)(u(x))
In this case, let's consider u(x) = x and v(x) = sin(2/x).
First, let's find the derivative of u(x):
(d/dx)(u(x)) = (d/dx)(x) = 1
Next, let's find the derivative of v(x). Since v(x) involves the composition of functions, we need to use the chain rule:
Let w = 2/x
Then, v(x) = sin(w)
Using the chain rule, the derivative of v(x) with respect to w is:
(d/dw)(v(x)) = (d/dw)(sin(w)) = cos(w)
Now, let's find the derivative of w with respect to x:
(d/dx)(w) = (d/dx)(2/x) = -2/x^2
Finally, putting it all together:
(d/dx)(y) = u(x)*(d/dx)(v(x)) + v(x)*(d/dx)(u(x))
= x*cos(w)*(-2/x^2) + sin(w)*1
= -2cos(w)/x + sin(w)
Substituting w = 2/x, we have:
(d/dx)(y) = -2cos(2/x)/x + sin(2/x)
This is the derivative of y = x*sin(2/x) with respect to x.