Question

Find the derivative of the function. $$ g(x)=\sqrt[6]{x} $$

Short answer

The derivative of the function \(g(x) = \sqrt[6]{x}\) is \(g'(x) = \frac{1}{6} *x^{-5/6}\).

Step-by-step solution

Convert to Exponential Form

Let's write the function \(g(x)\) in exponent form. Since \(\sqrt[6]{x}\) is the same as \(x^{1/6}\), our function becomes \(g(x) = x^{1/6}\).

Apply Power Rule

Applying the power rule, which states that the derivative of \(x^n\) is \(n*x^{n-1}\), the derivative of \(g(x)\) can be found by multiplying the exponent by the function and then decreasing the exponent by 1. Therefore, \(g'(x) = \frac{1}{6} *x^{(1/6)-1}\).

Simplify the Result

By simplifying the exponent, the result is \(-\frac{1}{6}x^{-5/6}\).

Key concepts

Power Rule

When finding the derivative of a function, you'll often encounter powers of x. The power rule is a fundamental tool in calculus that simplifies this process. To use the power rule, you look at functions in the form of f(x) = x^n, where n is any real number. The derivative, denoted as f'(x) or df/dx, is then determined by multiplying the exponent n with the function itself and subtracting 1 from the exponent. In formula terms, the rule can be expressed as:
\[f'(x) = n \times x^{n-1}\].
This is exactly what we needed to do in the given exercise to find the derivative. The process applies whether n is positive, negative, or even a fraction, as in the case of the given function g(x). By applying this rule, complex calculations can be broken down into straightforward steps that lead to the derivative of a power function.

Exponential Form

Converting to exponential form often serves as a preliminary step in calculus for handling roots and rational exponents. Roots like \(\root{n}{x} = x^{1/n}\) are rewritten in exponential form to leverage the power rule when finding derivatives. For example, the sixth root of x, which is \(\root{6}{x}\), is equivalent to x raised to the power of 1/6 when expressed in exponential form.
That's because the radical sign indicates a root, which can always be translated into a fractional exponent. The denominator of the fraction represents the root, and the numerator indicates the power, which for roots is typically one, hence the fraction 1/n. This was the initial step in our solution: converting the function g(x) to x^{1/6}, setting the stage for the application of the power rule.

Simplifying Exponent

After applying the power rule, the resulting exponent may need to be simplified. Simplifying an exponent means expressing the power in the most straightforward form possible, often to facilitate further calculations or simply to adhere to standard notation. In the case of the derivative we found in the exercise, we ended up with an exponent of (1/6) - 1, which simplifies to -5/6. This simplification is necessary for understanding the behavior of the function and preparing it for graphing, integrating, or other operations.
To explain, when you subtract one from a fraction like 1/6, you are effectively finding a common denominator with 1, which would be 6/6, and then subtracting: 1/6 - 6/6 = -5/6. This negative exponent is crucial as it represents the reciprocal function x^{-n} = 1/x^n, which in the context of our derivative, indicates a decreasing rate as x increases, signifying the inverse relationship between the function and its derivative.