Find the derivative of the following function. y=e^(-6x^(2)+3x) Answer: y^(')=

Identify function: Identify the function to differentiate. y = e^(-6x^2 + 3x)Recognize composition: Recognize that this is a composition of functions: the exponential function and a quadratic function. We will use the chain rule to differentiate. The outer function is e^u, where u = -6x^2 + 3x. The inner function is u(x) = -6x^2 + 3x.Derivative of outer function: Find the derivative of the outer function with respect to u. If v = e^u, then v' = e^u * du/dx.Derivative of inner function: Find the derivative of the inner function with respect to x. u'(x) = d/dx(-6x^2 + 3x) = d/dx(-6x^2) + d/dx(3x) = -12x + 3.Apply chain rule: Apply the chain rule: y' = v' * u'(x). y' = e^(-6x^2 + 3x) * (-12x + 3).Simplify derivative: Simplify the expression for the derivative. y' = (-12x + 3) * e^(-6x^2 + 3x).

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Q. Find the derivative of the following function.

\newline

y=e^{-6 x^{2}+3 x}

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Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $y = e^{(-6x^2 + 3x)}$
Recognize composition: Recognize that this is a composition of functions: the exponential function and a quadratic function. We will use the chain rule to differentiate. $\newline$ The outer function is $e^u$ , where $u = -6x^2 + 3x$ . $\newline$ The inner function is $u(x) = -6x^2 + 3x$ .
Derivative of outer function: Find the derivative of the outer function with respect to $u$ . If $v = e^u$ , then $v' = e^u \cdot \frac{du}{dx}$ .
Derivative of inner function: Find the derivative of the inner function with respect to $x$ . $u'(x) = \frac{d}{dx}(-6x^2 + 3x) = \frac{d}{dx}(-6x^2) + \frac{d}{dx}(3x) = -12x + 3$ .
Apply chain rule: Apply the chain rule: $y' = v' \cdot u'(x)$ . $y' = e^{(-6x^2 + 3x)} \cdot (-12x + 3)$ .
Simplify derivative: Simplify the expression for the derivative. $y' = (-12x + 3) \cdot e^{(-6x^2 + 3x)}.$