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To find the derivative \( y' \) of the function \( y = \frac{-7}{4 \sqrt[4]{x}} \), we need to rewrite the function in a more convenient form for differentiation and then apply the rules of differentiation.

First, rewrite the function:
\[ y = \frac{-7}{4 x^{1/4}} = -\frac{7}{4} x^{-1/4} \]

Now we differentiate \( y \) with respect to \( x \):
\[ y' = -\frac{7}{4} \cdot \frac{d}{dx} \left( x^{-1/4} \right) \]

Using the power rule \( \frac{d}{dx} x^n = n x^{n-1} \):
\[ \frac{d}{dx} x^{-1/4} = -\frac{1}{4} x^{-1/4 - 1} = -\frac{1}{4} x^{-5/4} \]

Therefore,
\[ y' = -\frac{7}{4} \cdot -\frac{1}{4} x^{-5/4} = \frac{7}{16} x^{-5/4} \]

So, the derivative \( y' \) is:
\[ y' = \frac{7}{16} x^{-5/4} \]

If you have any questions or need further details, please let me know.

Here are 5 related questions you might want to ask next:
1. How do you apply the chain rule for differentiation?
2. Can you explain the product rule with an example?
3. How do you find the derivative of a trigonometric function?
4. What is the process for finding the second derivative of a function?
5. How do you differentiate implicit functions?

**Tip:** When dealing with fractional exponents, it's often helpful to rewrite them in a form that makes differentiation straightforward using power rules.

Learn how to find the derivative of the function y = -7/(4 * x^(1/4)) with a detailed step-by-step solution using differentiation rules and power rules. This problem covers differentiation of functions involving fractional exponents, suitable for advanced high school or college students.

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