Find dy/dx if y=csc(2x−−√) . (1 point) Responses dydx=−csc(2x√)cot(2x√)x√ d y d x = − csc ( 2 x ) cot ( 2 x ) x dydx=csc(2x√)cot(2x√)x√ d y d x = csc ( 2 x ) cot ( 2 x ) x dydx=−2csc(2x√)cot(2x√)x√ d y d x = − 2 csc ( 2 x ) cot ( 2 x ) x dydx=2csc(2x√)cot(2x√)x√ d y d x = 2 csc ( 2 x ) cot ( 2 x ) x dydx=−csc(2x√)cot(2x√)2x√
The correct response is:
dy/dx = -2csc(2x√)cot(2x√)√
To find dy/dx for y = csc(2x-√), we can use the chain rule.
Let u = 2x-√
Now, we can rewrite the function as y = csc(u)
The derivative of csc(u) with respect to u is -csc(u)cot(u).
Now, let's find the derivative of u with respect to x.
du/dx = d(2x-√)/dx
= 2-√ * d(2x)/dx
= 2-√ * 2
= 4-√
Finally, we can apply the chain rule:
dy/dx = (dy/du) * (du/dx)
= -csc(u) * cot(u) * 4-√
= -4-√ * csc(2x-√) * cot(2x-√)
Therefore, the answer is: dy/dx = -4-√ * csc(2x-√) * cot(2x-√)
To find dy/dx (the derivative of y with respect to x) for the given function y = csc(2x-√), we can use the chain rule. The chain rule states that if f(g(x)) is a composite function, then its derivative is given by the product of the derivative of the outer function times the derivative of the inner function.
First, let's find the derivative of the outer function. The derivative of csc(u) is -csc(u)cot(u), where u represents the input of the csc function.
Next, let's differentiate the inner function. The derivative of 2x-√ is simply 2.
Now, we can apply the chain rule. The derivative of y = csc(2x-√) is given by:
dy/dx = d(csc(2x-√))/d(2x-√) * d(2x-√)/dx
Using the derivatives we found earlier, we have:
dy/dx = -csc(2x-√)cot(2x-√) * 2
Therefore, the correct answer is: dy/dx = -2csc(2x-√)cot(2x-√).