Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. $$ f(x)=4 \sqrt{x} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 4 \sqrt{x}\) is \(f'(x) = \frac{2}{\sqrt{x}}\).

Step 1: Rewrite the function

Rewrite the given function \(f(x) = 4 \sqrt{x}\) in the form of \(f(x) = ax^n\). Therefore, the function becomes \(f(x) = 4x^{1/2}\).

Step 2: Apply the power rule

Use the power rule, which states that \(d/dx[x^n] = nx^{n-1}\) to find the derivative. The rule is applied to every term of the function. Here \(a = 4, n = \frac{1}{2}\). Applying the power rule to our function will result \(f'(x) = \frac{1}{2} \cdot 4x^{(\frac{1}{2}-1)}\).

Step 3: Simplify the result

Simplify the expression to get the final answer, \(f'(x) = 2x^{-\frac{1}{2}}\). But remembering - a negative exponent means that the term is on the denominator. Therefore, the function can also be written as \(f'(x) = \frac{2}{x^\frac{1}{2}}\). This means for any x in the domain of \(f\), the derivative of the function \(f\) at \(x\) is \(f'(x) = \frac{2}{\sqrt{x}}\).