Use The Definition To Calculate The Derivative Of The Following Function. Then Find The Values Of The Derivative As Specified.$ P(\theta)=\sqrt{5 \theta} }$Calculate The Derivatives $[ P^{\prime (1), \quad P^{\prime}(5), \quad

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on calculating the derivative of a given function and finding its values at specific points.

The Function

The given function is:

${ p(\theta) = \sqrt{5 \theta} }$

This function represents a square root function with a coefficient of 5 and a variable θ.

Calculating the Derivative

To calculate the derivative of the function, we will use the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1).

In this case, we can rewrite the function as:

${ p(\theta) = (5 \theta)^{\frac{1}{2}} }$

Using the power rule, we can differentiate the function as follows:

${ p^{\prime}(\theta) = \frac{1}{2} (5 \theta)^{-\frac{1}{2}} \cdot 5 }$

Simplifying the expression, we get:

${ p^{\prime}(\theta) = \frac{5}{2 \sqrt{5 \theta}} }$

Finding the Values of the Derivative

Now that we have the derivative of the function, we can find its values at specific points.

Finding the Value of the Derivative at θ = 1

To find the value of the derivative at θ = 1, we substitute θ = 1 into the derivative:

${ p^{\prime}(1) = \frac{5}{2 \sqrt{5 \cdot 1}} }$

Simplifying the expression, we get:

${ p^{\prime}(1) = \frac{5}{2 \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot 5^{\frac{1}{2}}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot 5^{\frac{1}{2}}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

${ p^{\prime}(1) = \frac{5}{2 \cdot \sqrt{5}} }$

[ p^{\prime}(1