Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 18 , find the derivative of the given function. $$ f(x)=(3 x+5)^{2 / 3} $$

Step-by-Step Solution

Answer

The derivative of the function is f'(x) = 2(3x + 5)^{-1/3}.

Step 1 - Identify the function and its form

The given function is f(x) = (3x + 5)^{2/3}. This is a composition of functions, specifically an outer function g(u) = u^{2/3} and an inner function h(x) = 3x + 5.

Step 2 - Use the Chain Rule

To find the derivative of a composite function, use the Chain Rule: \[ f'(x) = g'(h(x)) \times h'(x) \].

Step 3 - Differentiate the Outer Function

Differentiate the outer function g(u) = u^{2/3}.Using the power rule, \[ g'(u) = \frac{2}{3}u^{-1/3} \].

Step 4 - Differentiate the Inner Function

Differentiate the inner function h(x) = 3x + 5.Using basic differentiation rules, \[ h'(x) = 3 \].

Step 5 - Apply the Chain Rule

Combine the results using the Chain Rule: \[ f'(x) = g'(h(x)) \times h'(x) \].Substitute the derivatives found:\[ f'(x) = \frac{2}{3}(3x + 5)^{-1/3} \times 3 \].

Step 6 - Simplify the Expression

Simplify the expression to get the final derivative:\[ f'(x) = 2(3x + 5)^{-1/3} \].

Key Concepts

Composite Function

A composite function combines two functions where the output of one function becomes the input for the next. In this exercise, it's essential to recognize that \(f(x) = (3x + 5)^{2/3}\) is a composite function. The outer function is \(g(u) = u^{2/3}\) and the inner function is \(h(x) = 3x + 5\). To differentiate composite functions properly, we use the chain rule.

Power Rule

When differentiating functions raised to a power, the power rule can simplify our work. For the function \(g(u) = u^{2/3}\), the derivative is found using the power rule formula \[ g'(u) = \frac{d}{du}(u^n) = nu^{n-1} \]. Applying this rule, we get \[ g'(u) = \frac{2}{3}u^{-1/3} \]. This step is crucial for solving our initial problem as it helps simplify the outer function.

Basic Differentiation

Basic differentiation rules help us find the derivative of simple functions. In the inner function \(h(x) = 3x + 5\), we apply the rule that states the derivative of a linear function \[ \frac{d}{dx}(ax + b) = a \]. Hence, \[ h'(x) = 3 \]. Combining this with the power rule via the chain rule provides the solution for the composite function.

• First, differentiate the outer function \[ g(u) = \frac{2}{3}u^{-1/3} \].
• Then, differentiate the inner function \[ h(x) = 3 \].

Utilizing the chain rule: \[ f'(x) = g'(h(x)) \times h'(x) = \frac{2}{3}(3x + 5)^{-1/3} \times 3 = 2(3x + 5)^{-1/3} \].