Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(x)=\sqrt[3]{12 x-5} $$

Short answer

The derivative of the function \(f(x) = \sqrt[3]{12x - 5}\) is \(f'(x) = \frac{4}{(12x - 5)^{2/3}}\).

Step-by-step solution

Rewrite the Function

Rewrite the function \(f(x)\) in the form that can be more easily differentiated. In this case, it's better to write it in its power form: \(f(x) = (12x - 5)^{1/3}\).

Apply the Chain Rule

Apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here the outer function is \(u^{1/3}\) and the inner function is \(u=12x - 5\). So we have: \(f'(x) = \frac{1}{3}(12x - 5)^{-2/3} \cdot (12)\).

Simplify the Result

Now let's simplify the given expression: \(f'(x) = \frac{12}{3}(12x - 5)^{-2/3} = 4(12x - 5)^{-2/3}\). This expression simplifies to: \(f'(x) = \frac{4}{(12x - 5)^{2/3}}\).

Key concepts

Chain Rule

The chain rule is a fundamental tool in calculus used for finding the derivative of a composite function. A composite function is a function made up of two functions. For example, if you have a function inside another function, you'll need to use the chain rule to differentiate it properly.
The chain rule states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function. In notation, if you have a function like \(f(g(x))\), the derivative is given by \(f'(g(x)) \, \cdot\, g'(x)\). This means you first find the derivative of the outer function while keeping the inner function the same, and then multiply it by the derivative of the inner function.
Applying the chain rule helps when functions are layered and ensures the transition between functions is accounted for in the calculation of the derivative.

Power Rule

The power rule is one of the simplest and most frequently used rules in differentiation. It is used when dealing with functions of the form \(x^n\). According to the power rule, if you have \(f(x) = x^n\), then the derivative is \(f'(x) = n \cdot x^{n-1}\).
This rule works by lowering the power by one and multiplying by the original power, simplifying the process of differentiation significantly.

• For example, if you have \(f(x) = x^3\), then the derivative is \(f'(x) = 3x^2\).
• The power rule is applicable for both positive and negative exponents.
• It's very handy for quickly finding derivatives of polynomials and power-based functions.

In our original equation \(f(x) = (12x - 5)^{1/3}\), the power rule is used in combination with the chain rule to find the derivative.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of the function with respect to a variable. In simpler terms, it tells you how much the function's output changes when you change the input slightly.

• Differentiation helps in understanding how a function behaves, such as whether it's increasing or decreasing.
• Derivatives have numerous applications, from physics (to find velocity and acceleration) to economics (for cost and revenue analysis).
• It involves the application of rules like the sum rule, product rule, and others, depending on the structure of the function.

Finding the derivative involves algebraic manipulations and applying different rules of calculus. In this exercise, differentiation involves rewriting the function to a power form, using the power and chain rules to find the derivative.

Composite Function

A composite function is created when one function is applied to the result of another function. In the context of calculus, understanding composite functions is crucial, especially when it comes to differentiation.
A composite function is expressed as \(f(g(x))\), meaning function \(f\) is applied to the result of \(g(x)\). In our case, the function \(f(x) = \sqrt[3]{12x-5}\) or equivalently \(f(x) = (12x-5)^{1/3}\) is composite because it applies the power function to the linear expression \(12x-5\).

• The chain rule is necessary for differentiating these kinds of functions, as it involves multiple layers.
• Composites are commonly encountered in real-world problems where processes are dependent on other processes.

Understanding how to work with composite functions is essential for effective calculus problem-solving. It allows you to break down complex tasks into simpler, manageable parts.