Problem 23 - University Calculus: Early Transcendentals

Question

Find the derivatives of the functions in Exercises \(23-50\). $$p=\sqrt{3-t}$$

Step-by-Step Solution

Answer

The derivative is \(-\frac{1}{2}(3-t)^{-1/2}\)."

Step 1: Apply differentiation rules

Use power, product, quotient, or chain rule as appropriate.

Step 2: Simplify

The derivative is The derivative is \(-\frac{1}{2}(3-t)^{-1/2}\).".

Key Concepts

Power Function

A power function is a type of function that can be represented in the form \(f(x) = x^n\), where \(n\) is a real number. In our exercise, we deal with the function \(p = \sqrt{3-t}\). This can be rewritten using power notation as \((3-t)^{1/2}\). Rewriting square roots using exponents is crucial in calculus, as it allows you to use differentiation rules applicable to any power of \(x\). When you have a power function, you can apply the power rule of differentiation, which states that for any function \(f(x) = x^n\), the derivative \(f'(x) = nx^{n-1}\). This rule is a simple yet powerful tool that helps when finding the rate of change of functions expressed in the form of powers.

Chain Rule

The chain rule is fundamental when working with composite functions— those that can be written in the form \(f(g(x))\). It tells you how to differentiate such functions by taking the derivative of the outer function and multiplying it by the derivative of the inner function. When applying the chain rule to the function \((3-t)^{1/2}\), you consider \(u = 3-t\) as the inner function, and \(u^{1/2}\) as the outer function. To find the derivative, you first differentiate the outer function: if \(u^{1/2}\), then its derivative is \((1/2)u^{-1/2}\). Next, you find the derivative of the inner function, \(-1\) (since the derivative of \(3-t\) with respect to \(t\) is \(-1\)). Finally, combine these using the chain rule: \(\frac{d}{dt}[(3-t)^{1/2}] = (1/2)(3-t)^{-1/2} \times (-1)\). This step-by-step process highlights the importance of recognizing composite functions to apply derivation methods accurately.

Square Root Function

Square root functions are a special type of radical function. They are often expressed in the form \(f(x) = \sqrt{x}\) and can be found frequently in many mathematical contexts. The square root function \(\sqrt{3-t}\) represents a smooth curve which is defined for values of \(t\) less than or equal to 3. In calculus, converting square root functions into power functions, such as \((3-t)^{1/2}\), allows us to leverage the tools of differentiation, like the power and chain rules, to find derivatives effectively. Such transformations are vital for analyzing how quickly quantities change in response to variations in \(t\). Understanding how to work with square root functions and translate them into an exponent form is essential for effective calculus problem-solving.