Find StartFraction dy Over dx EndFraction
dy
dx.
yequals=RootIndex 4 StartRoot StartFraction 4 x Over 4 x minus 3 EndFraction EndRoot4
4x
4x−3

We are given the equation:

\[
y = \sqrt[4]{\frac{4x}{4x - 3}}
\]

Our goal is to find the derivative of \(y\) with respect to \(x\), i.e., \(\frac{dy}{dx}\).

### Step 1: Rewrite the equation in exponential form
Instead of the fourth root, we can express the equation using fractional exponents:

\[
y = \left( \frac{4x}{4x - 3} \right)^{\frac{1}{4}}
\]

### Step 2: Use the chain rule
To differentiate this expression, we'll use the chain rule. Let’s define the inner function as:

\[
u = \frac{4x}{4x - 3}
\]

Thus, \(y = u^{\frac{1}{4}}\).

Now, the chain rule gives:

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]

### Step 3: Differentiate \(y = u^{\frac{1}{4}}\)
The derivative of \(y\) with respect to \(u\) is:

\[
\frac{dy}{du} = \frac{1}{4} u^{-\frac{3}{4}}
\]

### Step 4: Differentiate \(u = \frac{4x}{4x - 3}\)
To differentiate \(u\) with respect to \(x\), we apply the quotient rule. The quotient rule states:

\[
\frac{du}{dx} = \frac{(4x - 3)(4) - (4x)(4)}{(4x - 3)^2}
\]

Simplifying the numerator:

\[
(4x - 3)(4) - (4x)(4) = 16x - 12 - 16x = -12
\]

Thus, we have:

\[
\frac{du}{dx} = \frac{-12}{(4x - 3)^2}
\]

### Step 5: Combine the results
Now, substituting everything back, we get:

\[
\frac{dy}{dx} = \frac{1}{4} u^{-\frac{3}{4}} \cdot \frac{-12}{(4x - 3)^2}
\]

Since \(u = \frac{4x}{4x - 3}\), substitute \(u\) back into the equation:

\[
\frac{dy}{dx} = \frac{1}{4} \left( \frac{4x}{4x - 3} \right)^{-\frac{3}{4}} \cdot \frac{-12}{(4x - 3)^2}
\]

Simplifying:

\[
\frac{dy}{dx} = \frac{-3}{(4x - 3)^2} \left( \frac{4x}{4x - 3} \right)^{-\frac{3}{4}}
\]

This is the derivative of \(y = \sqrt[4]{\frac{4x}{4x - 3}}\).

Would you like any details about the individual steps, or have further questions?

Here are 5 related questions to extend your understanding:
1. How does the chain rule apply to functions with fractional exponents?
2. What is the quotient rule, and when should it be used?
3. How do you simplify expressions involving negative exponents?
4. How would the derivative change if the root were cubic instead of the fourth root?
5. Can this function be extended to higher-order derivatives? If so, how?

**Tip:** When using the quotient rule, always differentiate the numerator and denominator separately before combining them into the final expression.

Find StartFraction dy Over dx EndFraction dy dx. y = RootIndex 4 StartRoot StartFraction 4x Over 4x - 3 EndFraction EndRoot

Learn how to find the derivative of the function y = RootIndex 4 StartRoot 4x / (4x - 3) EndRoot using the chain rule and quotient rule. This detailed step-by-step solution explains how to rewrite the function in exponential form, apply the chain rule, and use the quotient rule to simplify the derivative. Suitable for advanced calculus students in Grades 11-12.

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