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Find the derivative of the function y = 8e^x3 using the chain rule.

May 19, 2026Mathematics

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Answer

24x^2 e^{x^3}

Graph of y = 8e^(x³)

Key points of y = 8e^(x³)
Point	x	y
y-intercept	0	8

4 steps

The given function is $y = 8e^{x^3}$ . To find the derivative $\frac{dy}{dx}$ , we will use the chain rule.

Identify the outer and inner functions.
Let $u = x^3$ . Then the function becomes $y = 8e^u$ .
Differentiate the outer function with respect to $u$ and the inner function with respect to $x$ .
The derivative of $y = 8e^u$ with respect to $u$ is $\frac{dy}{du} = 8e^u$ .
The derivative of $u = x^3$ with respect to $x$ is $\frac{du}{dx} = 3x^2$ .
Apply the chain rule, which states $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .
$\frac{dy}{dx} = (8e^u) \cdot (3x^2)$
Substitute $u = x^3$ back into the expression and simplify.
$\frac{dy}{dx} = 8e^{x^3} \cdot 3x^2$ $\frac{dy}{dx} = 24x^2 e^{x^3}$
The derivative of the function is:
$24x^2 e^{x^3}$

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