Find the derivative of the following function. y=e^(-x^(3)) Answer: y^(')=

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Find the derivative of the following function.  y=e^(-x^(3)) Answer:  y^(')=

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Identify Function Components: Identify the function and its components for differentiation. The function y = e^(-x^(3)) can be seen as an outer function e^u where u = -x^(3).Derivative of Outer Function: Determine the derivative of the outer function with respect to u, where u = -x^(3). The derivative of e^u with respect to u is e^u. So, (d/du)(e^u) = e^u.Derivative of Inner Function: Find the derivative of the inner function u with respect to x, where u = -x^(3). The derivative of -x^(3) with respect to x is -3x^(2). So, (d/dx)(-x^(3)) = -3x^(2).Apply Chain Rule: Apply the chain rule to differentiate the composite function y = e^(-x^(3)). The chain rule states that (dy/dx) = (dy/du) * (du/dx). We have (dy/du) = e^u and (du/dx) = -3x^(2). Therefore, (dy/dx) = e^u * (-3x^(2)).Substitute and Simplify: Substitute u back into the derivative to get the final answer. Since u = -x^(3), we replace u with -x^(3) in the derivative. So, (dy/dx) = e^(-x^(3)) * (-3x^(2)).Simplify the expression to get the final derivative. y' = -3x^(2) * e^(-x^(3)). This is the derivative of the function y = e^(-x^(3)).

Find the derivative of the following function. $\newline$ $y=e^{-x^{3}}$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function. $\newline$ $y=e^{-x^{3}}$ $\newline$ Answer: $y^{\prime}=$