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Find the derivative of the following function.

y=e^(-3x^(3)-7x^(2))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=e^{-3 x^{3}-7 x^{2}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=e^{-3 x^{3}-7 x^{2}}

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ The function given is $y = e^{(-3x^3 - 7x^2)}$ . We need to find the derivative of this function 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 $e^u$ and the inner function is $u = -3x^3 - 7x^2$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function $u$ . The derivative of $e^u$ with respect to $u$ is $e^u$ . So, the derivative of the outer function is $e^{(-3x^3 - 7x^2)}$ .
Differentiate inner function: Differentiate the inner function $u = -3x^3 - 7x^2$ with respect to $x$ . The derivative of $-3x^3$ with respect to $x$ is $-9x^2$ , and the derivative of $-7x^2$ with respect to $x$ is $-14x$ . Therefore, the derivative of the inner function is $-9x^2 - 14x$ .
Apply chain rule multiplication: Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. $\newline$ The derivative of the outer function is $e^{(-3x^3 - 7x^2)}$ , and the derivative of the inner function is $-9x^2 - 14x$ . Multiplying these together gives us the derivative of $y$ with respect to $x$ . $\newline$ $y' = e^{(-3x^3 - 7x^2)} \times (-9x^2 - 14x)$
Simplify derivative: Simplify the expression for the derivative. $\newline$ $y' = -9x^2 \cdot e^{(-3x^3 - 7x^2)} - 14x \cdot e^{(-3x^3 - 7x^2)}$ $\newline$ This is the final form of the derivative.

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