Problem 29 - Calculus Single Variable

Question

Use the method of Example 1 to calculate \(f^{\prime}(x)\) for the given function \(f\). $$ f(x)=\sqrt{3 x+7} $$

Step-by-Step Solution

Answer

The derivative of the function is \(f'(x) = \frac{3}{2\sqrt{3x + 7}}\).

Step 1: Identify the Outer and Inner Functions

The function given is a composite function. Identify that the outer function is the square root function, and the inner function is the linear function: \(g(x) = 3x + 7\) and \(h(u) = \sqrt{u}\), where \(u = 3x + 7\).

Step 2: Differentiate the Outer Function

Find the derivative of the outer function \(h(u) = \sqrt{u}\). The derivative is \(h'(u) = \frac{1}{2\sqrt{u}}\).

Step 3: Differentiate the Inner Function

Differentiate the inner function \(g(x) = 3x + 7\). The derivative is \(g'(x) = 3\).

Step 4: Apply the Chain Rule

Apply the chain rule, which states that if \(f(x) = h(g(x))\), then \(f'(x) = h'(g(x)) \cdot g'(x)\). Substitute the derivatives found in steps 2 and 3: \(f'(x) = \frac{1}{2\sqrt{3x + 7}} \cdot 3\).

Step 5: Simplify the Expression

Simplify the expression for \(f'(x)\): \(f'(x) = \frac{3}{2\sqrt{3x + 7}}\).

Key Concepts

Differentiation

In calculus, differentiation is the method of finding the derivative of a function. Essentially, it measures how a function changes as its input changes. Understanding this concept is crucial in tackling problems related to rates of change, slopes of curves, and motion.During differentiation, certain rules and functions need to be understood and applied:

• The Power Rule: Useful for differentiating terms like \(x^n\), where this results in \(nx^{n-1}\).
• The Constant Rule: The derivative of a constant is always zero.
• The Sum/Difference Rule: The derivative of a sum (or difference) of functions is simply the sum (or difference) of their derivatives.

When you differentiate functions, especially composite ones, having a clear understanding of how these rules apply individually and together is key. For composite functions, you'll need to apply the chain rule, which is a bit different from these basic rules.

Composite Functions

Composite functions involve combining two or more functions into a single function. It’s like nesting functions within each other. If you have a function \(f(x)\) and another function \(g(x)\), forming \(f(g(x))\), you create a composite function.Understanding the components of composite functions means identifying the inner function and the outer function. In formal terms, for \(f(g(x))\):

• The inner function is \(g(x)\).
• The outer function is \(f(u)\), where \(u = g(x)\).

This nesting is essential in problems requiring the chain rule because the rule helps differentiate composite functions by handling the inner and outer functions independently. Recognizing which functions play these roles in a composite function is a crucial step in utilizing the chain rule effectively.

Derivative of Square Root Function

The derivative of a square root function involves some specific rules and transformations, specifically when the function is embedded in a composite setting.For a basic square root function, \(f(x) = \sqrt{x}\), its derivative can be found using the chain rule because it can be rewritten using a power: \(f(x) = x^{1/2}\). Differentiating this using the power rule results in:\[f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}\]In the context of composite functions like \(\sqrt{3x + 7}\), the derivative must first identify the inner function \(3x + 7\) and apply the chain rule. This involves finding the derivative of the outer function first (the square root) and then multiplying by the derivative of the inner function. This combination allows you to approach more complex functions systematically, achieving not just the derivative of simple expressions, but of much more involved ones.