Given the function y=sin(sqrt(6-4x^(2))), find (dy)/(dx).

Question

Given the function  y=sin(sqrt(6-4x^(2))), find  (dy)/(dx).

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Identify outermost function: Identify the outermost function and start differentiating using the chain rule. The outermost function is the sine function. We will apply the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Differentiate outer function: Differentiate the outer function with respect to its inner function. The derivative of sin(u) with respect to u is cos(u), where u is the inner function sqrt(6-4x^2). So we have: (dy)/(dx) = cos(sqrt(6-4x^2)) * (d(sqrt(6-4x^2)))/(dx)Differentiate inner function: Differentiate the inner function sqrt(6-4x^2) with respect to x. The inner function is a square root function, and we can rewrite sqrt(6-4x^2) as (6-4x^2)^(1/2). Using the chain rule again, the derivative of (6-4x^2)^(1/2) with respect to x is (1/2)(6-4x^2)^(-1/2) * (d(6-4x^2))/(dx).Differentiate innermost function: Differentiate the innermost function 6-4x^2 with respect to x. The derivative of 6 with respect to x is 0, and the derivative of -4x^2 with respect to x is -8x. So we have: (d(6-4x^2))/(dx) = -8xCombine derivatives: Combine the derivatives from the previous steps to find the final derivative. Substitute the derivative of the innermost function into the derivative of the inner function, and then multiply by the derivative of the outer function: (dy)/(dx) = cos(sqrt(6-4x^2)) * (1/2)(6-4x^2)^(-1/2) * (-8x)Simplify expression: Simplify the expression. (dy)/(dx) = -4x * cos(sqrt(6-4x^2)) / sqrt(6-4x^2) This is the simplified form of the derivative.

Given the function $\newline$ $y=\sin(\sqrt{6-4x^{2}}),$ find $\newline$ $(\frac{dy}{dx}).$

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Given the function\newliney=sin⁡(6−4x2),y=\sin(\sqrt{6-4x^{2}}),y=sin(6−4x2​), find \newline(dydx).(\frac{dy}{dx}).(dxdy​).

Given the function $\newline$ $y=\sin(\sqrt{6-4x^{2}}),$ find $\newline$ $(\frac{dy}{dx}).$