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Find the derivative of the following function.

y=e^(-6x^(6)-9x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=e^{-6 x^{6}-9 x^{5}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=e^{-6 x^{6}-9 x^{5}}

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ We are given the function $y = e^{(-6x^6 - 9x^5)}$ . We need to find its derivative with respect to $x$ .
Apply chain rule: Apply the chain rule for differentiation. $\newline$ 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. In this case, the outer function is $e^{u}$ , where $u = -6x^{6} - 9x^{5}$ , and the inner function is $u(x) = -6x^{6} - 9x^{5}$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function. The derivative of $e^u$ with respect to $u$ is $e^u$ . So, the derivative of $e^{(-6x^6 - 9x^5)}$ with respect to the inner function $u$ is $e^{(-6x^6 - 9x^5)}$ .
Differentiate inner function: Differentiate the inner function with respect to $x$ . The inner function $u(x) = -6x^6 - 9x^5$ is a polynomial, and we differentiate it term by term. The derivative of $-6x^6$ with respect to $x$ is $-36x^5$ , and the derivative of $-9x^5$ with respect to $x$ is $-45x^4$ .
Combine using chain rule: Combine the results using the chain rule. $\newline$ Multiplying the derivative of the outer function by the derivative of the inner function, we get: $\newline$ $y' = e^{(-6x^6 - 9x^5)} \cdot (-36x^5 - 45x^4)$
Simplify expression: Simplify the expression if possible. $\newline$ In this case, there is no further simplification needed, so the final answer is: $\newline$ $y' = e^{(-6x^6 - 9x^5)} \cdot (-36x^5 - 45x^4)$

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