Calculate the derivative of the following functions.
y = ⁴√(2x / (4x - 3))

Verified step by step guidance

1. Step 1: Rewrite the function in a more convenient form for differentiation. The function y = \sqrt[4]{\frac{2x}{4x - 3}} can be rewritten using exponent notation as y = \left(\frac{2x}{4x - 3}\right)^{1/4}.
2. Step 2: Apply the chain rule for differentiation. The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) \cdot g'(x). Here, let u = \frac{2x}{4x - 3}, so y = u^{1/4}.
3. Step 3: Differentiate the outer function with respect to u. The derivative of u^{1/4} with respect to u is \frac{1}{4}u^{-3/4}.
4. Step 4: Differentiate the inner function u = \frac{2x}{4x - 3} with respect to x. Use the quotient rule, which states that if u = \frac{v}{w}, then u' = \frac{v'w - vw'}{w^2}. Here, v = 2x and w = 4x - 3.
5. Step 5: Combine the results from Steps 3 and 4 using the chain rule. Multiply the derivative of the outer function by the derivative of the inner function to find the derivative of the original function y with respect to x.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative can be calculated using various rules, such as the power rule, product rule, quotient rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions that involve roots or powers, as seen in the given function.

Quotient Rule

The quotient rule is used to find the derivative of a function that is the ratio of two other functions. It states that if you have a function defined as the quotient of two functions, the derivative is given by the formula: (f/g)' = (f'g - fg') / g², where f and g are the numerator and denominator functions, respectively. This rule is essential for differentiating the given function, which involves a fraction.