Question

In Exercises \(31-42,\) find \(d y / d x\). $$y=(1-6 x)^{2 / 3}$$

Short answer

The derivative of the function \(y = (1 - 6x)^{2 / 3}\) with respect to x is \(-4(1 - 6x)^{-1 / 3}\)

Step-by-step solution

Identify the Outer and Inner Functions

Prepare to apply the chain rule, \(d / dx[f(g(x))] = f'(g(x)) * g'(x)\). In our case, the outer function is \(f(u) = u^{2 / 3}\) and the inner function is \(g(x) = 1 - 6x\).

Compute the Derivative of the Outer Function

Find the derivative of the outer function \(f(u) = u^{2 / 3}\) using the power rule which gives \(f'(u) = (2 / 3)u^{-1 / 3}\).

Differentiate the Inner Function

Differentiate the inner function \(g(x) = 1 - 6x\). The derivative \(g'(x)\) is equal to -6.

Apply the Chain Rule

Calculate the derivative of the entire function by applying the chain rule: substitute \(g(x)\) and \(g'(x)\) into \(f'(g(x)) * g'(x)\). This results in \((2 / 3)(1 - 6x)^{-1 / 3} * -6\).

Simplify the Result

Rewrite \((2 / 3)(1 - 6x)^{-1 / 3} * -6\) in a simple and clear way. The final result is is \(-4(1 - 6x)^{-1 / 3}\).

Key concepts

Derivative Calculation

Understanding the process of finding a derivative, which is a fundamental concept in calculus, is like uncovering the instantaneous rate of change of a function at any given point. Think of it as the speedometer reading of a car at a precise moment, showing exactly how fast you're going.

The derivative of a function can be calculated using several rules and theorems, and often, functions are compositions of other functions. This is where the chain rule comes in handy. It is an essential technique that deals with the derivative of a composed function. If you picture a function within a function, the chain rule helps by unwrapping each layer to reveal the overall rate of change. It informs us how to differentiate a complex function by breaking it down into its inner and outer parts. The exercise given demonstrates this beautifully by calculating the derivative of a composed function using the chain rule.

Power Rule

A tool that simplifies derivative calculation is known as the power rule. In the power rule, for any function of the form \(f(x) = x^n\), where \(n\) is any real number, the derivative of that function, noted as \(f'(x)\), is \(nx^{n-1}\). It's a straightforward and quick method to differentiate functions where the variable \(x\) is raised to a power.

The exercise shows the power rule in action by taking the derivative of an outer function raising \(x\) to the power of \(\frac{2}{3}\). Just apply the rule by bringing down the exponent as a coefficient and subtracting one from the exponent. It's essential to remember this rule, as it's frequently employed in differentiability and function analysis.

Function Differentiation

Differentiation, if viewed as an analytical tool, allows us to dissect functions to find the instantaneous rate of change. It can be seen as a surgical procedure where differentiation tools, such as the power rule, product rule, and quotient rule, are the scalpel.

With function differentiation, not only do you grasp how a function is behaving at any point, but you also understand the nuances of its growth and decay. It's much like understanding the mood swings of the function—knowing where it's increasing or decreasing, and where it's at its happiest or saddest (which mathematicians refer to as maxima and minima). The process described in the exercise is a classic case of function differentiation by employing the chain rule. This highlights the interconnectivity of various differentiation techniques when dealing with more complex functions.