Find the derivative of the following function. y=e^(x^(3)+4x^(2)) Answer: y^(')=

Identify Function Components: Identify the function and its components. We have y = e^(x^3 + 4x^2). The outer function is e^u, where u is the inner function u(x) = x^3 + 4x^2.Derivative of Outer Function: Find the derivative of the outer function with respect to u. The derivative of e^u with respect to u is e^u.Derivative of Inner Function: Find the derivative of the inner function u(x) = x^3 + 4x^2 with respect to x. Using the power rule, the derivative of x^3 is 3x^2, and the derivative of 4x^2 is 8x. So, u'(x) = d/dx(x^3 + 4x^2) = 3x^2 + 8x.Apply Chain Rule: Apply the chain rule to find the derivative of y with respect to x. The chain rule 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. So, y' = (d/du(e^u)) * (d/dx(u(x))) = e^(x^3 + 4x^2) * (3x^2 + 8x).

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Q. Find the derivative of the following function.

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y=e^{x^{3}+4 x^{2}}

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Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have $y = e^{x^3 + 4x^2}$ . The outer function is $e^u$ , where $u$ is the inner function $u(x) = x^3 + 4x^2$ .
Derivative of Outer Function: Find the derivative of the outer function with respect to $u$ . The derivative of $e^u$ with respect to $u$ is $e^u$ .
Derivative of Inner Function: Find the derivative of the inner function $u(x) = x^3 + 4x^2$ with respect to $x$ . Using the power rule, the derivative of $x^3$ is $3x^2$ , and the derivative of $4x^2$ is $8x$ . So, $u'(x) = \frac{d}{dx}(x^3 + 4x^2) = 3x^2 + 8x$ .
Apply Chain Rule: Apply the chain rule to find the derivative of $y$ with respect to $x$ . The chain rule 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. So, $y' = \left(\frac{d}{du}(e^u)\right) \cdot \left(\frac{d}{dx}(u(x))\right) = e^{x^3 + 4x^2} \cdot (3x^2 + 8x)$ .