Find the derivative of the following function. y=e^(-3x^(5)-6x^(4)) Answer: y^(')=

Identify Function: Identify the function to differentiate. We have y = e^(-3x^(5)-6x^(4)). This is an exponential function with a composite exponent.Apply Chain Rule: Apply the chain rule for differentiation. 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. Let u(x) = -3x^(5)-6x^(4), then y = e^(u(x)).Differentiate Outer Function: Differentiate the outer function with respect to u. The outer function is e^(u), and its derivative with respect to u is e^(u). (dy/du) = e^(u)Differentiate Inner Function: Differentiate the inner function u(x) with respect to x. u(x) = -3x^(5)-6x^(4) u'(x) = d/dx(-3x^(5)) + d/dx(-6x^(4)) u'(x) = -15x^(4) - 24x^(3)Apply Chain Rule: Apply the chain rule to find the derivative of y with respect to x. y' = (dy/du) * (du/dx) y' = e^(-3x^(5)-6x^(4)) * (-15x^(4) - 24x^(3))

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Q. Find the derivative of the following function.

\newline

y=e^{-3 x^{5}-6 x^{4}}

\newline

Answer:

y^{\prime}=

Identify Function: Identify the function to differentiate. $\newline$ We have $y = e^{(-3x^{5}-6x^{4})}$ . This is an exponential function with a composite exponent.
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. $\newline$ Let $u(x) = -3x^{5}-6x^{4}$ , then $y = e^{u(x)}$ .
Differentiate Outer Function: Differentiate the outer function with respect to $u$ . The outer function is $e^{u}$ , and its derivative with respect to $u$ is $e^{u}$ . $\frac{dy}{du} = e^{u}$
Differentiate Inner Function: Differentiate the inner function $u(x)$ with respect to $x$ .
$u(x) = -3x^{5}-6x^{4}$
$u'(x) = \frac{d}{dx}(-3x^{5}) + \frac{d}{dx}(-6x^{4})$
$u'(x) = -15x^{4} - 24x^{3}$
Apply Chain Rule: Apply the chain rule to find the derivative of $y$ with respect to $x$ . $y' = \frac{dy}{du} \cdot \frac{du}{dx}$ $y' = e^{(-3x^{5}-6x^{4})} \cdot (-15x^{4} - 24x^{3})$