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The power rule is a fundamental concept in differentiation that makes finding derivatives of polynomial functions straightforward. If you have a function of the form \( f(x) = x^n \), the power rule states that its derivative, denoted as \( f'(x) \), is \( nx^{n-1} \). This rule is incredibly useful because it allows us to quickly see how a change in \( x \) affects the function.To apply the power rule, you simply multiply the exponent \( n \) by the function's base \( x \), and then subtract one from the exponent. This process gives us the slope of the tangent line to the curve at any point. For example, if \( f(x) = x^3 \), its derivative would be \( f'(x) = 3x^2 \). This tells us that the slope changes with \( x \), specifically increasing as \( x \).

When we talk about derivatives, we're referring to a way of determining how a function's output changes as its input changes. In the exercise, we're asked to differentiate the function \( f(x) = \frac{2}{3}x^{\frac{3}{2}} \). To find the derivative, \( f'(x) \), we zero in on \( x^{\frac{3}{2}} \) while keeping the constant factor \( \frac{2}{3} \) outside. Using the power rule, multiply the exponent \( \frac{3}{2} \) by the function's constant:

• The exponent: \( \frac{3}{2} \)
• Multiply by constant: \( \frac{2}{3} \times \frac{3}{2} = 1 \)

The derivative now becomes \( f'(x) = x^{\frac{3}{2} - 1} = x^{\frac{1}{2}} \), simplifying the problem tremendously. This derivative shows the rate of change of the function as \( x \) changes.

Simplifying derivatives is a key step in making the result as clear and direct as possible. Starting with our derivative \( f'(x) = \frac{2}{3}\left(\frac{3}{2}x^{\frac{1}{2}}\right) \), we first multiply \( \frac{2}{3} \) by \( \frac{3}{2} \), which equals 1, simplifying to \( f'(x) = x^{\frac{1}{2}} \).Through simplification, expressions become easier to understand and work with for further mathematical manipulation. In our case, expressing \( x^{\frac{1}{2}} \) as \( \sqrt{x} \) makes the output more familiar since square roots are commonly encountered. Hence, the final simplified derivative is \( f'(x) = \sqrt{x} \), which reflects how the function changes with respect to \( x \). Remember, a simplified result is not just about aesthetics, but also about providing clarity and insight into the function's behavior.