Question

Prove (Theorem 2.3) that \(\frac{d}{d x}\left[x^{n}\right]=n x^{n-1}\) for the case in which \(n\) is a rational number. (Hint: Write \(y=x^{p / q}\) in the form \(y^{q}=x^{p}\) and differentiate implicitly. Assume that \(p\) and \(q\) are integers, where \(q>0 . )\)

Short answer

The derivative of \(x^{n}\) where \(n\) is a rational number is \(n x^{n-1}\). This is obtained by transforming the original equation, differentiating implicitly, and doing the algebra to isolate the derivative.

Step-by-step solution

Rewrite the Function

Rewrite \(y = x^{p/q}\) in the form \(y^{q} = x^{p}\). This manipulation allows us to differentiate both sides implicitly.

Differentiate Implicitly

Differentiating both sides with respect to \(x\) gives \( qy^{q-1} * y' = px^{p-1} \), where \(y'\) is the derivative of \(y\) with respect to \(x\).

Isolate y'

Isolate \(y'\) to get the derivative, which gives \(y' = px^{p-1} / (qy^{q-1})\).

Substitute y back

Now substitute \(y = x^{p/q}\) back into the equation to get \(y' = (px^{p-1}) / (qx^{p-q})\). After simplification, this results in \(y' = p/q * x^{p/q - 1}\). After canceling out the terms, this results in \(y' = n x^{n-1}\)

Key concepts

Implicit Differentiation

Implicit differentiation is a powerful technique used to find the derivative of an equation that is not explicitly solved for one variable in terms of another. This approach is particularly useful when dealing with functions that involve variables raised to a certain power, like in the case of rational exponents.

For example, when dealing with an equation like \(y^q = x^p\), where \(pq\) are integers, you cannot directly apply the typical rules of differentiation without first expressing \(y\) explicitly in terms of \(x\). However, implicit differentiation allows us to find \(\frac{dy}{dx}\) by taking the derivative of both sides of the equation with respect to \(x\), keeping in mind the chain rule. This involves differentiating \(y\) as if it were a function of \(x\) and then multiplying by the derivative of \(y\) itself, denoted as \(y'\).

Through implicit differentiation, we find that if \(y^q = x^p\), then \( qy^{q-1} * y' = px^{p-1}\), which is essential in solving for \(y'\) and proving that the power rule is valid even when the exponent is a rational number.

Power Rule for Derivatives

The power rule for derivatives is an essential rule in calculus that simplifies the process of finding the derivative of a power function. When the function is presented in the form \(x^n\), where \(n\) is a real number, the power rule states that the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\).

This rule simplifies the differentiation process because it eliminates the need for more cumbersome limit processes to find slopes of tangent lines or rates of change. It is particularly useful because it applies to both integer and rational exponents, confirming its broad applicability across functions. When applied to rational exponents, it allows students to express derivatives in a more simplified form, making the understanding and computation of the derivatives much more approachable.

Derivative of Exponential Functions

Exponential functions, which can generally be denoted as \(a^x\) where \(a\) is a positive constant, have unique properties when it comes to differentiation. However, our focus here is on functions with bases that are variables, such as \(x^p\), with rational exponents.

The process of finding the derivative of exponential functions with a variable base and a rational exponent involves transforming the function into a form that can utilize implicit differentiation. As demonstrated in the exercise, by writing the function as \(y = x^{p/q}\) and then re-arranging to \(y^q = x^p\), and differentiating implicitly, we arrive at an expression for the derivative that mirrors the properties of the derivative of a standard exponential function. This approach highlights how implicit differentiation seamlessly extends to more complex functions such as rational exponentials.

Rational Exponents

Rational exponents represent the connection between exponentiation and root operations. A rational exponent, such as \(p/q\), indicates that the base should be raised to the power of \(p\) and also that a \(q\)-th root should be taken. In the context of derivatives, functions of the form \(x^{p/q}\) can pose a challenge.

However, as seen in the provided solution, by applying the concept of rewriting the function with the rational exponent in a form that allows for implicit differentiation, we facilitate the use of the power rule. The rewriting process allows us to convert a function with a rational exponent into a simpler form that implicitly reflects the existence of both the power and the root. Undoubtedly, understanding rational exponents is crucial for comprehending their behavior under differentiation which, as proved, follows the power rule.