Problem 20 - Calculus an Applied Approach

Question

Find the derivative of the function. $$ g(x)=4 \sqrt[3]{x}+2 $$

Step-by-Step Solution

Answer

The derivative of the function is \( g'(x) = 4/3*x^{-2/3}\)

Step 1: Recognize the function form

The function \( g(x)=4 \sqrt[3]{x}+2 \) can be written as \( g(x)=4x^{1/3}+2 \). The exponent \(1/3\) indicates that this is a cube root function, and we have to use the power rule to differentiate it.

Step 2: Apply the power rule

The power rule states that for any real number \(n\), the derivative of \(x^n\) is \(nx^{n-1}\). Applying the power rule to \(4x^{1/3}\), we obtain: \(1/3*4x^{1/3-1} = 4/3*x^{-2/3}\). The constant 2 derives to 0, as the derivative of a constant is always 0.

Step 3: Write the final answer

Putting it all together, the derivative of the function is: \( g'(x) = 4/3*x^{-2/3}\)

Key Concepts

Power rule

The power rule is a simple yet powerful technique used for finding derivatives of polynomial functions. Imagine you have a function of the form \(x^n\), where \(n\) can be any real number. Applying the power rule allows us to find the derivative quickly. The rule says: to differentiate \(x^n\), multiply by the exponent \(n\), and then decrease the exponent by 1. For example, if our function is \(x^3\), its derivative according to the power rule will be \(3x^{2}\). Similarly, for a fractional or negative exponent, the process is the same. If you have \(x^{-2}\), its derivative becomes \(-2x^{-3}\).

This method simplifies differentiation, especially when dealing with functions that have terms like \(4x^{1/3}\), as in our original problem. In this case, the power rule helps us find derivatives of non-integer powers smoothly, providing a foundation for more complex calculus operations.

Cube root function

A cube root function is a specific type of radical function where the function \(g(x) = \sqrt[3]{x}\) can also be expressed in terms of exponents as \(g(x) = x^{1/3}\). This transformation into an exponent form is crucial because it allows us to apply the power rule for differentiation.

Understanding cube root functions is essential because they are common in various mathematical contexts, especially when examining real-world scenarios involving volumes, where dimensional scaling laws might apply. By expressing the cube root function in exponential form, calculations become more straightforward, as regular differentiation techniques apply.

For instance, in our exercise, changing \(\sqrt[3]{x}\) to \(x^{1/3}\) helps leverage the power rule to find derivatives more efficiently. This boosts accuracy and reduces computational complexities, making solving such mathematical problems much easier.

Constant derivative

Constants play a unique role in calculus, especially in differentiation. Differentiating a constant value is one of the simplest operations in calculus. The derivative of a constant is always 0. This rule arises because a constant function is flat and has no change; therefore, its rate of change is zero.

For example, if you have \(g(x) = c\), where \(c\) is a constant, the derivative \(g'(x)\) will be 0. This property holds true regardless of whether the constant is positive, negative, or zero.

• Effectively, dealing with constants can simplify calculations when finding derivatives of more elaborate functions.
• In the original exercise, the constant is represented by 2 in \(g(x) = 4 \sqrt[3]{x} + 2\). Differentiating it yields 0, allowing us to focus solely on differentiating the variable part \(4x^{1/3}\).
• This simplification forms an essential step in solving calculus problems efficiently.

Understanding this concept ensures you can identify and eliminate constants effortlessly during differentiation.