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Find the derivative of the following function.

y=e^(-6x^(3)+5x^(2))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=e^{-6 x^{3}+5 x^{2}}$ $\newline$ 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^{-6 x^{3}+5 x^{2}}

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ $y = e^{-6x^{3} + 5x^{2}}$ $\newline$ We need to find the derivative of $y$ with respect to $x$ , denoted as $y'$ .
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. $\newline$ In this case, the outer function is $e^{u}$ and the inner function is $u = -6x^{3} + 5x^{2}$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function $u$ . If $y = e^u$ , then the derivative of $y$ with respect to $u$ is $\frac{dy}{du} = e^u$ .
Differentiate inner function: Differentiate the inner function $u$ with respect to $x$ .
$u = -6x^{3} + 5x^{2}$
The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = \frac{d}{dx}(-6x^{3}) + \frac{d}{dx}(5x^{2})$
$\frac{du}{dx} = -18x^{2} + 10x$
Apply chain rule: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ $y' = \frac{dy}{du} \cdot \frac{du}{dx}$ $\newline$ $y' = e^{(-6x^{3} + 5x^{2})} \cdot (-18x^{2} + 10x)$

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