Problem 7 - Thomas Calculus

Question

Find \(d y / d x\) in Exercises \(1-10\) $$ y=(2 x+5)^{-1 / 2} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = -(2x+5)^{-3/2} \).

Step 1: Identify the Function

The function given is \( y = (2x+5)^{-1/2} \), which is a composition of the functions \( f(u) = u^{-1/2} \) and \( u = 2x + 5 \).

Step 2: Apply The Chain Rule

To differentiate the function, use the chain rule. The chain rule states that if a function \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Step 3: Differentiate the Outer Function

The outer function is \( f(u) = u^{-1/2} \). Differentiating this with respect to \( u \) gives \( f'(u) = -\frac{1}{2}u^{-3/2} \).

Step 4: Differentiate the Inner Function

The inner function is \( u = 2x + 5 \). The derivative of \( u \) with respect to \( x \) is \( g'(x) = 2 \).

Step 5: Combine Derivatives Using the Chain Rule

Combine the derivatives according to the chain rule: \( \frac{dy}{dx} = f'(u) \cdot g'(x) = -\frac{1}{2}(2x+5)^{-3/2} \cdot 2 \).

Step 6: Simplify the Result

Simplify the result: \( \frac{dy}{dx} = -\frac{1}{2}(2x+5)^{-3/2} \cdot 2 = -(2x+5)^{-3/2} \).

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus used for differentiating composite functions. A composite function is when you have one function nested inside another. Think of it as stepping through layers of an onion; differentiation here involves peeling back each layer systematically. When you apply the chain rule, you're essentially taking the derivative of the outer function and multiplying it by the derivative of the inner function. For example, if a function is of the form \(y = f(g(x))\), according to the chain rule, the derivative is \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\). This approach simplifies complex differentiation problems, making it easier to handle functions that are compositions of simpler ones. In our exercise, we started with \(y = (2x+5)^{-1/2}\). The two components at play are the outer function \(f(u) = u^{-1/2}\) and the inner function \(u = 2x + 5\). By applying the chain rule to this setup, you're essentially breaking down the problem into more manageable parts.

Basics of Derivatives

Derivatives represent the rate of change of a function with respect to a variable, and they are a primary tool in calculus for understanding how functions behave. Essentially, the derivative tells you how a function's output value changes as its input value changes slightly. For any function, say \(y = f(x)\), the derivative \(\frac{dy}{dx}\) represents how \(y\) changes with a small change in \(x\). This concept is visually represented as the slope of the tangent line to the curve at any given point. In our given exercise, the outer function \(f(u) = u^{-1/2}\) involves differentiating a power function. Using the power rule for differentiation, \(f(u) = u^n\) differentiates to \(f'(u) = nu^{n-1}\). Applying this to \(u^{-1/2}\), we get \(f'(u) = -\frac{1}{2}u^{-3/2}\). This shows how derivatives can be systematically calculated using rules learned in calculus.

Calculus Problem-Solving Strategies

Solving calculus problems often involves breaking down a problem into simpler parts and applying fundamental rules. This approach makes it easier to manage even complex problems systematically. Here are some strategies:

• Identify the type of function: Determine if the function is simple, composite, or involves additional rules like the product or quotient rule.
• Choose the right differentiation method: Depending on the function type, decide whether to use direct differentiation, chain rule, product rule, etc.
• Calculate step-by-step: By applying the differentiation method step-by-step, you're less likely to make mistakes. You can check each part before moving on to the next.
• Simplify expressions: Finally, algebraic manipulation can be necessary to simplify the derivative expression to make it easy to understand and use.

In our step-by-step exercise solution, we apply these strategies to find \(\frac{dy}{dx} = -(2x+5)^{-3/2}\). By breaking the problem down—first identifying the functions involved, then correctly applying the chain rule, and simplifying at the end—we systematically arrive at the solution.