Problem 13 - Calculus - AP Edition

Question

$$\text { Find } f^{\prime}(x)$$ $$f(x)=x^{e}+\frac{1}{x^{\sqrt{10}}}$$

Step-by-Step Solution

Answer

\(f'(x) = e \, x^{e-1} - \sqrt{10} \, x^{-\sqrt{10}-1}\)

Step 1: Differentiate the first term

To differentiate the function, start by finding the derivative of the first term, which is \(x^{e}\). According to the power rule, the derivative of \(x^a\) is \(a \, x^{a-1}\). Here, \(a = e\), so the derivative is \(e \, x^{e-1}\).

Step 2: Differentiate the second term

Next, find the derivative of the second term, \(\frac{1}{x^{\sqrt{10}}}\). Rewrite this as \(x^{-\sqrt{10}}\). Again, apply the power rule. The derivative is \(-\sqrt{10} \, x^{-\sqrt{10}-1}\).

Step 3: Combine the derivatives

Now that we have the derivatives of both terms, combine them to find the overall derivative of the function. The derivative of \(f(x)\) is \(f'(x) = e \, x^{e-1} - \sqrt{10} \, x^{-\sqrt{10}-1}\).

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus, especially when it comes to differentiation. It's a straightforward method used to calculate the derivative of power functions, which are expressions of the form \(x^n\), where \(n\) is any real number.
The power rule states that if you have a function \(f(x) = x^n\), then its derivative \(f'(x)\) is given by bringing down the exponent as a coefficient and reducing the exponent by one, which results in \(n \, x^{n-1}\).

• For example, to differentiate \(x^3\), you apply the rule to get the derivative \(3 \, x^{2}\).
• This principle also applies to non-integer exponents like \(e\), as seen in our problem, where \(f(x) = x^e\) leads to \(f'(x) = e \, x^{e-1}\).

The power rule offers an efficient way to differentiate many algebraic expressions and is a critical tool in a calculus toolbox.

Derivative of Polynomials

In calculus, a polynomial is a mathematical expression that consists of variables raised to whole number exponents and constant coefficients. Polynomial expressions can range from simple monomials to complex combinations of terms.
The derivative of a polynomial function is found by differentiating each term individually. Thanks to the linearity of differentiation, each term can be handled separately and then summed up to form the final derivative.

• Take, for example, \(f(x) = 3x^2 + 4x + 5\). The derivative, \(f'(x)\), is the sum of the derivatives of each term: \(6x + 4\).
• The process involves applying the power rule to each term where the exponent is a positive integer.

The derivative of polynomials is always a polynomial of one degree less, making the technique useful for understanding the behavior and slope of polynomial curves.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent. These functions exhibit rapid growth or decay. However, when differentiating exponential functions with a variable in the exponent, additional rules apply beyond the basic power rule.
In scenarios involving expressions such as \(f(x) = a^{x}\), where \(a\) is a positive constant, the derivative is found using principles of natural logarithms. However, when the base of an exponential function is the mathematical constant \(e\), the differentiation process simplifies greatly because \(e^x\) differentiates to \(e^x\).

• In our initial problem, we encounter the expression \(x^e\) where \(x\) is the base, and \(e\) is the exponent. Since the exponent is constant, the power rule works effectively here.
• If the scenario involved \(e^x\), the persistent rate of change means its derivative is itself: \(e^x\).

Understanding how to differentiate exponential functions is essential in calculus due to their common occurrence in mathematical modeling.