Questions: Use the definition to calculate the derivative of the following function. Then find the values of the derivative as specified. p(θ)=√(5θ); p′(1), p′(5), p′(2/5) p′(θ)=□

Solution Steps

To find the derivative of the function \( p(\theta) = \sqrt{5\theta} \), we can use the power rule for differentiation. The power rule states that if \( f(\theta) = \theta^n \), then \( f'(\theta) = n\theta^{n-1} \). We will rewrite \( \sqrt{5\theta} \) as \( (5\theta)^{1/2} \) and apply the chain rule. After finding the general derivative, we will evaluate it at the specified points: \( \theta = 1 \), \( \theta = 5 \), and \( \theta = \frac{2}{5} \).

Given the function \( p(\theta) = \sqrt{5\theta} \), we can rewrite it as: \[ p(\theta) = \sqrt{5} \cdot \sqrt{\theta} = \sqrt{5} \cdot \theta^{1/2} \]

To find the derivative \( p'(\theta) \), we use the power rule: \[ \frac{d}{d\theta} \left( \theta^{1/2} \right) = \frac{1}{2} \theta^{-1/2} \] Thus, \[ p'(\theta) = \sqrt{5} \cdot \frac{1}{2} \theta^{-1/2} = \frac{\sqrt{5}}{2\sqrt{\theta}} \]

Evaluate \( p'(\theta) \) at \( \theta = 1 \): \[ p'(1) = \frac{\sqrt{5}}{2\sqrt{1}} = \frac{\sqrt{5}}{2} \]

Evaluate \( p'(\theta) \) at \( \theta = 5 \): \[ p'(5) = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2} \]

Evaluate \( p'(\theta) \) at \( \theta = \frac{2}{5} \): \[ p'\left(\frac{2}{5}\right) = \frac{\sqrt{5}}{2\sqrt{\frac{2}{5}}} = \frac{\sqrt{5}}{2} \cdot \sqrt{\frac{5}{2}} = \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{\sqrt{2}} = \frac{5}{2\sqrt{2}} = \frac{5\sqrt{2}}{4} \approx 0.7906 \cdot \sqrt{5} \]

Final Answer