Question

Find the derivative of the function. $$ f(x)=\sqrt[5]{x} $$

Short answer

The derivative of the function \(f(x)=\sqrt[5]{x}\) is \(f'(x) = \frac{1}{5x^{4/5}}\).

Step-by-step solution

Rewrite the Function

Rewrite the function \(f(x)=\sqrt[5]{x}\) in exponential form. The function can be written as \(f(x)=x^{1/5}\).

Apply the Power Rule

Apply the power rule, which is \(d/dx[x^n] =n*x^{n-1}\). The derivative is thus \((1/5)*x^{1/5-1}\). This simplifies to \(f'(x)=(1/5)*x^{-4/5}\).

Rewrite in Radical Form

Convert the solution to radical form for simplicity and clearer understanding. Replacing \(-n\) with \(\frac{1}{n}\), we can rewrite the derivative as \(f'(x) = \frac{1}{5} \sqrt[5]{x^{-4}}\). This simplifies to \(f'(x) = \frac{1}{5x^{4/5}}\).

Key concepts

Power Rule for Derivatives

Understanding the power rule for derivatives is essential for anyone studying calculus. This rule is a fundamental shortcut that helps you find the derivative of a function where the variable, typically written as 'x', is raised to a power. To apply the power rule, for a function in the form of \( f(x) = x^n \), the derivative \( f'(x) \) is calculated as \( nx^{n-1} \).

For example, if you have \( f(x) = x^3 \), using the power rule, the derivative would be \( 3x^{3-1} = 3x^2 \). The simplicity of the power rule makes it one of the most commonly used techniques in differentiation. This rule is particularly useful because it applies to any real number 'n', which can even be a fraction, just like in the exercise provided, where the function \( f(x) = \( x^{1/5} \) \) is differentiated.

Exponential Form in Calculus

When you encounter a radical symbol in calculus, it's often beneficial to translate it into exponential form. This is because the operations pertaining to exponents are well-defined and can help simplify the process of differentiation. The general rule states that \( \sqrt[n]{x} = x^{1/n} \).

Applying this concept to the provided exercise, the radical function \( f(x)=\(sqrt{\)\( x^{5} \)\} \), which can also be considered a type of 'root' function, is translated into the function \( f(x) = x^{1/5} \). This transformation to exponential form prepares the function for more streamlined differentiation using rules like the power rule.

Simplifying Derivatives

Simplifying derivatives can often make the analysis of functions and the application of further calculus principles much more approachable. Simplifying the derivative entails reforming it so that it’s easier to interpret or apply to a problem. This includes using algebraic manipulation to combine like terms, factoring, and getting rid of negative exponents.

In the original problem, the derivative \( f'(x)=(1/5)*x^{-4/5} \) contains a negative exponent, which often indicates a reciprocal. To simplify, this can be rewritten as \( f'(x) = \frac{1}{5x^{4/5}} \). This form is typically easier to understand and work with, especially when evaluating the derivative at specific points.

Radical Notation in Calculus

Radical notation is another way to represent roots of numbers and functions in calculus. This notation is more familiar to most students from their prior education in basic algebra. Expressing derivatives in radical notation can sometimes make them seem less intimidating and can make evaluating them by hand more straightforward.

In calculus, converting from exponential form back to radical form can be helpful, as in the case where the derivative function \( f'(x) = (1/5)*x^{-4/5} \) is converted to \( f'(x) = \frac{1}{5} \(sqrt{\)\( x^{-4} \)\} = \frac{1}{5x^{4/5}} \). Though the expression has the same value, the use of radicals might be more intuitive for students when graphing or applying the result to further calculations.