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It looks like the OCR extracted part of the equation as "√2x⁴ + sec(5x)." The full equation appears to be:

\[
y = \sqrt[3]{2x^4 + \sec(5x)}
\]

Now, let's find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).

We'll use the chain rule, power rule, and the derivative of secant for this. The derivative of a cube root expression involves rewriting it as a power of \(1/3\).

I'll compute this derivative step-by-step.The derivative of the given function \( y = \sqrt[3]{2x^4 + \sec(5x)} \) with respect to \( x \) is:

\[
\frac{dy}{dx} = \left(2.67x^3 + 1.67 \cdot \tan(5x) \cdot \sec(5x)\right) \cdot \left(2x^4 + \sec(5x)\right)^{-2/3}
\]

Would you like a more detailed breakdown of the steps involved, or any further clarification on the result?

Here are some follow-up questions for you:
1. What rule governs the differentiation of powers and composite functions?
2. How does the chain rule apply to functions involving trigonometric components?
3. How do you differentiate secant and tangent functions?
4. What does the derivative of \( \sec(5x) \) with respect to \( x \) involve?
5. Why is the result simplified to involve powers of \(2/3\)?

**Tip:** Always rewrite cube roots as fractional exponents when differentiating.

Find the derivative of y = cube root of (2x^4 + sec(5x))

Learn how to find the derivative of the function y = cube root of (2x^4 + sec(5x)). This solution involves using the chain rule, power rule, and the derivative of the secant function, suitable for undergraduate calculus students.

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