Problem 10 - Calculus an Applied Approach

Question

find the second derivative of the function. $$ f(x)=x \sqrt[3]{x} $$

Step-by-Step Solution

Answer

The second derivative of the function \( f(x)=x \sqrt[3]{x} \) is \( f''(x) = \frac{4}{9}x^{-2/3} \).

Step 1: Simplify the function

The function \( f(x)=x \sqrt[3]{x} \) can be simplified by writing it as \[ f(x)=x \cdot x^{1/3}=x^{1+1/3}=x^{4/3} \]

Step 2: Find the first derivative

The rule for differentiation powers of \( x \) is \( \frac{d}{dx} x^{n} = n x^{n-1} \). Applying this rule, the first derivative of the function becomes: \[ f'(x)= \frac{4}{3}x^{4/3 - 1} = \frac{4}{3}x^{1/3} \]

Step 3: Find the second derivative

Once again apply the power rule, to find the second derivative: \[ f''(x) = \frac{1}{3} \cdot \frac{4}{3}x^{1/3-1} = \frac{4}{9}x^{-2/3} \]

Key Concepts

Power Rule in Differentiation

The power rule is a quick, efficient way to calculate derivatives of functions that involve powers of \( x \). It takes the form \( \frac{d}{dx} x^n = n x^{n-1} \). This means you take the exponent \( n \), bring it out front as a coefficient, and reduce the original exponent by one.

This technique is incredibly useful:

• When you have a simple power function like \( x^3 \), where applying the power rule gives \( 3x^{2} \).
• It's equally applicable to fractional powers, as seen in the exercise where \( x^{4/3} \) transitions through the power rule to \( \frac{4}{3}x^{1/3} \).
• Even negative and zero powers can be differentiated using the power rule.

Understanding and utilizing the power rule significantly simplifies the process of finding derivatives for polynomial and polynomial-like functions.

Understanding Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. A derivative represents the rate of change of a function's output with respect to its input, essentially describing how the function's value changes as \( x \) changes.

The purpose:

• To determine slopes of functions at any point.
• Find rates of change in real-world applications, such as speed in physics.
• Identify and analyze critical points where a function has maximum, minimum or saddle points.

For the function \( f(x) = x^{4/3} \), differentiation provides not only one rate of change but through multiple derivatives, deeper insights into its behavior across its curve.
With each differentiation step using rules like the power rule, you peel back a layer to explore further into the curve's dynamics. Differentiating once gives you the slope, but differentiating again gives you the concavity, indicating acceleration or deceleration of the change.

Simplification of Functions

Simplifying functions helps in making the differentiation process more manageable and less error-prone. By reducing complex expressions into simpler ones, not only does it make applying rules like the power rule easier, but it reduces potential calculation mistakes.

The function initially given was \( f(x) = x \sqrt[3]{x} \), where the cube root complicates straightforward differentiation.

• Through simplification, it transforms into \( x^{4/3} \), a basic power function.
• This allows for a seamless application of differentiation rules.
• Manages fractional exponents directly without dealing with roots.

By rewriting functions in simpler algebraic forms, whether transforming roots to fractional powers or combining like terms, you set yourself up for smoother differentiation steps and more accurate results.