Problem 45 - Calculus and its applications

Question

Find \(f^{\prime}(x)\) for the given function. \(f(x)=\sqrt{x}\) (See Example 8 in Section 1.3.)

Step-by-Step Solution

Answer

The derivative is \( f^{\prime}(x) = \frac{1}{2\sqrt{x}} \).

Step 1: Rewrite the function using exponent notation

The function is given as \( f(x) = \sqrt{x} \). To differentiate it easily, rewrite the square root as an exponent: \( f(x) = x^{1/2} \).

Step 2: Differentiate using the power rule

The power rule for differentiation states that if \( f(x) = x^n \), then \( f^{\prime}(x) = nx^{n-1} \). Here, \( n = 1/2 \). Substitute into the power rule: \( f^{\prime}(x) = \frac{1}{2}x^{(1/2)-1} = \frac{1}{2}x^{-1/2} \).

Step 3: Simplify the derivative

Convert the expression \( \frac{1}{2}x^{-1/2} \) back to a form with a square root to simplify it. This gives \( f^{\prime}(x) = \frac{1}{2\sqrt{x}} \).

Key Concepts

Power Rule

The power rule is an essential tool in calculus for differentiating polynomial functions. If you have a function in the form of \( f(x) = x^n \), where \( n \) is any real number, the derivative of that function is given by multiplying \( n \) by \( x \) raised to the power of \( n-1 \). So, the formula for the derivative is: \ - \( f^{\prime}(x) = nx^{n-1} \) \ Using this rule makes differentiating terms like \( x^{3} \) or \( x^{1/2} \) straightforward. Simply multiply the coefficient (which is often 1 when not explicitly shown) by the exponent, then reduce the exponent by one. Think of it as bringing down the exponent in front of the variable and reducing the "power" by one. This rule is especially helpful when the function is expressed using exponent notation, which is our next concept.

Derivative of Square Root

The derivative of a square root function involves transforming the square root into a form that is easier to handle algebraically. A square root like \( \sqrt{x} \) can be rewritten using exponent notation as \( x^{1/2} \). This transformation is critical as it allows us to apply the power rule.Here's how it works step by step:

• Rewrite the square root: Transform \(\sqrt{x}\) to \(x^{1/2}\).
• Apply the Power Rule: Use \( f^{\prime}(x) = \frac{1}{2}x^{(1/2)-1} \).
• Simplify: Simplify \( \frac{1}{2}x^{-1/2} \) back to \( \frac{1}{2\sqrt{x}} \).

This method efficiently helps to calculate the derivative and provides a clear path from start to finish for solving problems involving square roots in calculus.

Exponent Notation

Exponent notation, also known as power notation, is a method of expressing numbers using bases and exponents, which simplifies algebraic expressions. It is particularly useful when differentiating functions that include roots or fractional exponents.For example, \( x^{1/2} \) is another way to represent the square root of \( x \), \( \sqrt{x} \). This notation makes it easier to apply rules such as the power rule for differentiation. Why use exponent notation?

• Simplification: Converts complicated expressions into simpler, shorter forms.
• Ease of differentiation: Allows straightforward application of differentiation rules like the power rule.
• Flexibility: Accommodates both integer and fractional exponents.

By converting root expressions to exponent form, a seemingly complex differentiation problem becomes manageable and results in more streamlined calculations.