Find the derivative of the following function. y=e^(-8x^(4)) Answer: y^(')=

Identify Functions: Identify the inner function and the outer function. The inner function is u(x) = -8x^4, and the outer function is y = e^u, where u is the inner function.Derivative of Outer Function: Find the derivative of the outer function with respect to u. The derivative of e^u with respect to u is e^u. (dy/du) = e^uDerivative of Inner Function: Find the derivative of the inner function with respect to x. The derivative of -8x^4 with respect to x is -32x^3. (du/dx) = d/dx(-8x^4) = -32x^3Apply Chain Rule: Apply the chain rule to find the derivative of y with respect to x. The chain rule states that dy/dx = (dy/du) * (du/dx). (dy/dx) = e^u * (-32x^3)Substitute Inner Function: Substitute the inner function back into the derivative. Since u = -8x^4, we substitute it back into the expression for dy/dx. (dy/dx) = e^(-8x^4) * (-32x^3)Simplify Final Derivative: Simplify the expression to find the final derivative. y' = -32x^3 * e^(-8x^4)

Home

Math Problems

Calculus

Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=e^{-8 x^{4}}

\newline

Answer:

y^{\prime}=

Identify Functions: Identify the inner function and the outer function. $\newline$ The inner function is $u(x) = -8x^4$ , and the outer function is $y = e^u$ , where $u$ is the inner function.
Derivative of Outer Function: Find the derivative of the outer function with respect to $u$ . The derivative of $e^u$ with respect to $u$ is $e^u$ . $(dy/du) = e^u$
Derivative of Inner Function: Find the derivative of the inner function with respect to $x$ . The derivative of $-8x^4$ with respect to $x$ is $-32x^3$ . $\frac{du}{dx} = \frac{d}{dx}(-8x^4) = -32x^3$
Apply Chain Rule: Apply the chain rule to find the derivative of $y$ with respect to $x$ . The chain rule states that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ . $\frac{dy}{dx} = e^u \cdot (-32x^3)$
Substitute Inner Function: Substitute the inner function back into the derivative. $\newline$ Since $u = -8x^4$ , we substitute it back into the expression for $\frac{dy}{dx}$ . $\newline$ $\frac{dy}{dx} = e^{(-8x^4)} \cdot (-32x^3)$
Simplify Final Derivative: Simplify the expression to find the final derivative. $y' = -32x^3 e^{-8x^4}$