Problem 56 - Calculus an Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{\sqrt{x+2}} $$

Step-by-Step Solution

Answer

The derivative of the function \(y = \frac{1}{\sqrt{x + 2}}\) is \(y' = -\frac{1}{2\sqrt{(x + 2)^3}}\). The chain rule and the power rule were used for differentiation.

Step 1: Express the function using exponents

To start with, express the square root and the fraction in terms of exponents: \(y = (x + 2)^{-0.5}\)

Step 2: Apply the chain rule

Now, apply the chain rule, which states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is \(f(x) = x^{-0.5}\) and the inner function is \(g(x) = x + 2\). The derivative of \(y\) then is \(y' = -0.5(x + 2)^{-1.5} \cdot g'(x)\)

Step 3: Differentiate the inner function

Differentiate the inner function \(g(x) = x + 2\). As the derivative of a constant is zero and the derivative of \(x\) is 1, the differentiation gives \(g'(x) = 1\).

Step 4: Complete the chain rule

Complete the chain rule by replacing \(g'(x)\) in the equation for \(y'\). It becomes \(y' = -0.5(x + 2)^{-1.5}\)

Step 5: Simplify the function

Finally, express the negative exponent as a fraction and the exponent \(1.5\) as a square root. After simplifying, \(y' = -\frac{1}{2\sqrt{(x + 2)^3}}\)

Key Concepts

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. A composite function is one where a function is applied inside another function. For instance, with \( y = (x+2)^{-0.5} \), the function \((x + 2)\) is inside another function \(x^{-0.5}\).

• The chain rule helps to differentiate these types of functions by relating the derivative of the composite function to the derivative of its inner and outer functions.
• If you have \( h(x) = f(g(x)) \), the chain rule states that \( h'(x) = f'(g(x)) \cdot g'(x) \).
• In this exercise, \( f(x) = x^{-0.5} \) and \( g(x) = x+2 \). So, \( f'(g(x)) \) and \( g'(x) \) are calculated separately and then multiplied together for the final derivative.

Use of the chain rule simplifies the process of differentiation for nested functions. It breaks down the problem into manageable pieces.

Differentiation Rules

Differentiation rules help find the derivative, which shows the rate of change of a function. Several basic rules apply here, such as the power rule and the constant rule.

• The power rule states that if \( y = x^n \), then \( y' = nx^{n-1} \). This is used to differentiate terms involving powers of \(x\).
• The constant rule simply states that the derivative of a constant is zero. This comes in handy when differentiating terms like \( g(x) = x + 2 \).

For our function in the exercise, the differentiation rules assist in computing both the inner and outer derivatives needed for the chain rule. Understanding these basic rules is crucial for effectively applying advanced techniques like the chain rule.

Exponentiation

Exponentiation can make expressions involving roots or powers simpler to differentiate. Instead of using square roots, expressing them as exponents often simplifies the process.

• In the given function \( y = \frac{1}{\sqrt{x+2}} \), rewriting it as \( y = (x+2)^{-0.5} \) converts it into a form ready for differentiation.
• This step is critical because the power rule for differentiation requires the term to be in exponent form.

By re-expressing roots and fractions using exponents, differentiation rules like the power rule can be directly applied, making the process straightforward.

Function Composition

Function composition occurs when one function is applied to the result of another function. This is common in many mathematical expressions, including the one discussed in this problem.

• Understanding function composition is essential for applying the chain rule as it helps identify the inner and outer functions.
• For \( y = (x+2)^{-0.5} \), inside \( x^{-0.5} \), the \( x+2 \) acts as the inner function, illustrating function composition.

Recognizing and working with function composition is key to practicing differentiation, particularly with composite functions, making complex derivatives manageable.