Problem 56 - College Algebra and 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 \(dy/dx = -1/(2\sqrt{(x+2)^3})\). The chain rule and the power rule were used in this differentiation process.

Step 1: Rewrite the function

To simplify the differentiation process, rewrite the function from a division as a multiplication using negative exponent. Rewrite the function \(y=\frac{1}{\sqrt{x+2}}\) as \(y=(x+2)^{-1/2}\).

Step 2: Identify Outer and Inner Functions

In \(y=(x+2)^{-1/2}\), the outer function is \(f(x)=x^{-1/2}\) and the inner function is \(g(x)=x+2\).

Step 3: Apply the Chain Rule

The chain rule states that \(dy/dx = dy/du \cdot du/dx\), where \(u\) is the inner function \(g(x)\). Therefore, applying the chain rule, \(dy/dx = f'(g(x)) \cdot g'(x)\).

Step 4: Differentiate the Outer Function

For the outer function \(f(x)=x^{-1/2}\), differentiate using the power rule: \(f'(x) = -1/2 \cdot x^{-3/2} = -1/(2\sqrt{x^3})\). Place \(g(x) = x + 2\) into the derivative to get \(f'(g(x)) = -1/(2\sqrt{(x+2)^3})\).

Step 5: Differentiate the Inner Function

The derivative of \(g(x)=x+2\) is \(g'(x)=1\).

Step 6: Multiply the derivatives

Multiply \(f'(g(x))\) and \(g'(x)\) to find the derivative of the original function: \(dy/dx = f'(g(x)) \cdot g'(x) = -1/(2\sqrt{(x+2)^3}) \cdot 1= -1/(2\sqrt{(x+2)^3})\).