Find the derivative of the following function. y=e^(-6x^(6)+3x^(5)) Answer: y^(')=

Identify function: Identify the function to differentiate. y = e^(-6x^6 + 3x^5) This is an exponential function with a composite function as the exponent.Apply chain rule: Apply the chain rule for differentiation, 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. Let u(x) = -6x^6 + 3x^5 be the inner function. The outer function is e^u.Differentiate outer function: Differentiate the outer function with respect to u. If v = e^u, then v' = e^u * du/dx.Differentiate inner function: Differentiate the inner function u(x) = -6x^6 + 3x^5 with respect to x. u'(x) = d/dx(-6x^6) + d/dx(3x^5) u'(x) = -36x^5 + 15x^4Combine derivatives: Combine the derivatives using the chain rule. y' = v' * u'(x) y' = e^u * (-36x^5 + 15x^4) Substitute u back into the equation. y' = e^(-6x^6 + 3x^5) * (-36x^5 + 15x^4)

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Q. Find the derivative of the following function.

\newline

y=e^{-6 x^{6}+3 x^{5}}

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ $y = e^{(-6x^6 + 3x^5)}$ $\newline$ This is an exponential function with a composite function as the exponent.
Apply chain rule: Apply the chain rule for differentiation, 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. $\newline$ Let $u(x) = -6x^6 + 3x^5$ be the inner function. $\newline$ The outer function is $e^u$ .
Differentiate outer function: Differentiate the outer function with respect to $u$ . If $v = e^u$ , then $v' = e^u \cdot \frac{du}{dx}$ .
Differentiate inner function: Differentiate the inner function $u(x) = -6x^6 + 3x^5$ with respect to $x$ .
$u'(x) = \frac{d}{dx}(-6x^6) + \frac{d}{dx}(3x^5)$
$u'(x) = -36x^5 + 15x^4$
Combine derivatives: Combine the derivatives using the chain rule. $\newline$ $y' = v' \cdot u'(x)$ $\newline$ $y' = e^{u} \cdot (-36x^{5} + 15x^{4})$ $\newline$ Substitute $u$ back into the equation. $\newline$ $y' = e^{(-6x^{6} + 3x^{5})} \cdot (-36x^{5} + 15x^{4})$