Problem 31 - Calculus an Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ f(t)=\sqrt{t+1} $$

Step-by-Step Solution

Answer

The derivative of the function \( f(t)=\sqrt{t+1} \) is \( f'(t)=\frac{1}{2\sqrt{t + 1}} \).

Step 1: Convert the Function to a Suitable Form

First, note that the function \( f(t) = \sqrt{t + 1} \) can be rewritten as \( f(t) = (t + 1)^{1/2} \). The square root of an expression is the same as that expression raised to the power of 1/2.

Step 2: Apply the General Power Rule

Apply the general power rule, which states that the derivative of \( u(x)^n \) is \( n \cdot u(x)^{n - 1} \cdot u'(x) \). Here, \( u(x) = t + 1 \), \( n = 1/2 \), and \( u'(x) = 1 \) (since the derivative of \( t + 1 \) is 1). Applying the power rule yields \( f'(t) = \frac{1}{2} \cdot (t + 1)^{1/2 - 1} \cdot 1 \).

Step 3: Simplify the Equation

Simplify the equation to find the derivative. This simplifies to \( f'(t) = \frac{1}{2} \cdot (t + 1)^{-1/2} = \frac{1}{2\sqrt{t + 1}} \).

Key Concepts

Understanding the General Power Rule

The General Power Rule is a powerful tool for finding derivatives of functions that are raised to a power. This rule applies to expressions of the form \( u(x)^n \), where \( u(x) \) is a function and \( n \) is any real number. To differentiate such functions, follow this simple process:

• Identify \( n \), the power to which the function is raised.
• Find the derivative of \( u(x) \), denoted as \( u'(x) \).
• Apply the formula: \( n \cdot u(x)^{n-1} \cdot u'(x) \).

In the context of our exercise, \( f(t) = \sqrt{t+1} \), the expression is rewritten as \( (t+1)^{1/2} \). Here, \( n = 1/2 \) and \( u'(t) = 1 \), making the application of the rule straightforward.

Rewriting Functions for Easier Differentiation

Oftentimes, functions presented in their original formats can be trickier to differentiate. Rewriting functions into a more suitable form helps us to apply differentiation rules seamlessly. For instance, a square root function like \( \sqrt{t+1} \) is more manageable when expressed as \( (t+1)^{1/2} \).

This transformation is particularly useful when applying the power rule. Change roots or other complex expressions to exponent notation. This method not only simplifies the differentiation process but also minimizes common errors.

In every differentiation problem, examine the function to see if a rewrite is necessary. It often reduces complexities, allowing for a smoother solution.

Derivative Simplification: Making the Math Neat

After applying the General Power Rule, you often end up with expressions that need simplification. Simplification not only makes the math cleaner but also easier to interpret. In our case, starting with the derivative \( f'(t) = \frac{1}{2} \cdot (t+1)^{-1/2} \), we simplify it further.

To simplify:

• Change negative exponents into fractions.
• Rewrite the expression \( (t+1)^{-1/2} \) as \( \frac{1}{\sqrt{t+1}} \).

Combining these adjustments gives \( f'(t) = \frac{1}{2\sqrt{t+1}} \). This neater form is generally more acceptable in both mathematical solutions and real-world applications. Always aim for the most straightforward, clear final expression.