Problem 33 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=\left[\left(x^{2}+1\right)^{3}+x\right]^{3} $$

Step-by-Step Solution

Answer

The derivative is \(3\left(\left(x^{2}+1\right)^{3}+x\right)^{2} \cdot \left(3(x^{2}+1)^{2} \cdot 2x + 1\right)\).

Step 1: Identify the Outer Function

Recognize the structure of the function. The outermost function here is a cubic power: \[ u^{3} \]where \[ u = \left(x^{2}+1\right)^{3}+x \]

Step 2: Differentiate the Outer Function Using the Power Rule

Apply the power rule, which gives us:\[ \frac{d}{dx} \left( u^{3} \right) = 3u^{2} \cdot \frac{du}{dx} \]

Step 3: Identify and Differentiate the Inner Function

The inner function \[ u \] is \[ \left(x^{2}+1 \right)^{3}+x \]. Differentiate it:\[ \frac{du}{dx} = 3\left(x^{2}+1\right)^{2} \cdot 2x + 1 \]This is obtained using the chain rule for the \( \left(x^{2}+1 \right)^{3} \) and the derivative of \( x \).

Step 4: Combine to Find the Derivative of the Original Function

Substitute \( u \) with its original expression and \( \frac{du}{dx} \) into the result from Step 2:\[ \frac{d}{dx}\left( \left[\left(x^{2}+1\right)^{3}+x\right]^{3} \right) = 3\left(\left(x^{2}+1\right)^{3}+x\right)^{2} \cdot \left(3(x^{2}+1)^{2} \cdot 2x + 1\right) \]

Step 5: Simplify the Expression

Now simplify, if necessary, for further calculations or make this form ready for analysis. However, the expression derived gives a complete form of the derivative and represents the required solution.

Key Concepts

Understanding Derivatives

The concept of a derivative is fundamental in calculus. Derivatives help us understand how a function changes as its input changes. In practical terms, the derivative of a function gives us the function's rate of change or the slope of the function at any given point. In our exercise, we need to find the derivative of a function defined as:\[f(x) = ig((x^{2} + 1)^{3} + xig)^{3}\]The aim is to systematically apply rules of differentiation to find how this function changes with respect to \( x \). To achieve this, we break the problem down into smaller parts, focusing on both the inner and outer functions within the expression and applying differentiation rules accordingly.

Applying the Chain Rule

The chain rule is a powerful tool in differentiation. It is used when you need to differentiate a composition of functions. Essentially, the chain rule allows us to differentiate complex expressions by differentiating its pieces step-by-step. In our task, the chain rule comes into play while differentiating the inner function \( u = (x^{2}+1)^{3}+x \). During this process, we take note of how the parts of \( u \) interrelate, and apply the chain rule to account for the nested functions. The steps are:- Differentiate the outermost function, treating the inner function as a single variable.- Then, differentiate the inner function itself and multiply the derivatives together.This results in an expression where every layer of the function contributes to the final rate of change.

Utilizing the Power Rule

The power rule is a simple and powerful tool for differentiating expressions that involve powers of \( x \). It states that if \( f(x) = x^n \), then:\[\frac{d}{dx}[x^n] = nx^{n-1}\]In our exercise, we repeatedly utilize the power rule, both for:- Differentiating the outer function \( (u)^3 \)- As well as applying it to various parts of the inner function, such as \( (x^2+1)^3 \).For the outer function, differentiating \( u^3 \) involves using the power rule with \( u \) taken as a whole:- The derivative is \( 3u^{2} \), where the exponent 3 becomes a coefficient, and \( u \) is raised to the power of 2.This exemplifies how the power rule streamlines differentiation, making it easier to manage complex expressions like those in our exercise.