Problem 231 - Calculus Volume 1

Question

Find \(\frac{d y}{d x}\) for each function. $$ y=\left(2 x^{3}-x^{2}+6 x+1\right)^{3} $$

Step-by-Step Solution

Answer

The derivative is \( 3(2x^3 - x^2 + 6x + 1)^2 (6x^2 - 2x + 6) \).

Step 1: Identify the Rule to Use

The function is of the form \( [g(x)]^n \), where \( g(x) = 2x^3 - x^2 + 6x + 1 \) and \( n = 3 \). This suggests the use of the chain rule for differentiation.

Step 2: Apply the Chain Rule

The chain rule states that if \( y = [u(x)]^n \), then \( \frac{dy}{dx} = n[u(x)]^{n-1} \cdot \frac{du}{dx} \). Identify \( u(x) = 2x^3 - x^2 + 6x + 1 \), so \( y = [u(x)]^3 \).

Step 3: Differentiate the Inner Function

Differentiate \( u(x) = 2x^3 - x^2 + 6x + 1 \). The derivative is \( \frac{du}{dx} = 6x^2 - 2x + 6 \).

Step 4: Differentiate the Outer Function and Combine

Substitute \( n = 3 \), \( u(x) = 2x^3 - x^2 + 6x + 1 \), and \( \frac{du}{dx} = 6x^2 - 2x + 6 \) into the chain rule formula: \[\frac{dy}{dx} = 3(2x^3 - x^2 + 6x + 1)^2 \cdot (6x^2 - 2x + 6)\]

Step 5: Simplify the Expression

The expression above represents the derivative. No further simplification is necessary for the final expression:\[\frac{dy}{dx} = 3(2x^3 - x^2 + 6x + 1)^2 (6x^2 - 2x + 6)\]

Key Concepts

Differentiation

Differentiation is an essential concept in calculus that involves finding how a function changes as its input changes. It's all about rates of change. For instance, if you think about a car moving, differentiation helps you find the car's speed given the position at different times.

The key idea here is that differentiation calculates the derivative of a function, which represents the function's rate of change at any given point. Using differentiation, we can understand how rapidly something changes - whether it’s cost, temperature, position, or any measurable quantity.

In the context of the problem, differentiating the function encompassed applying a specific rule from calculus, known as the chain rule. This was necessary because the function we were dealing with was composed, which means it had one function nestled inside another.

Calculus

Calculus is the branch of mathematics focused on change. It’s divided into two main areas: differentiation and integration. Differentiation deals with finding the derivative, which is the rate of change, as discussed earlier, while integration is about finding the total or the whole.

In calculus, problems often involve rates and accumulations. This might be calculating the speed of a car or finding the area under a curve (which tells you something like total distance traveled). These problems require specific techniques and formulas.

• **Differentiation:** For finding how quantities change.
• **Integration:** For finding overall quantities given a rate of change.

For the exercise, knowledge of calculus allowed us to apply the chain rule effectively. This is a clever method of tackling functions that aren't straightforward, ensuring precise computations for derivatives.

Derivative of Polynomial Functions

The derivative of polynomial functions is a fundamental concept within differentiation. Polynomials are algebraic expressions made up of terms, which consist of constants, variables, and positive integer exponents.

To find their derivatives, we apply basic rules of differentiation:

• For any term like \( ax^n \), the derivative is \( nax^{n-1} \).
• This rule allows us to "reduce" the power by one and multiply by the original power to get the derivative.

In our problem, the inner function \( u(x) = 2x^3 - x^2 + 6x + 1 \) is a polynomial. So, we differentiated it term by term to get \( \frac{du}{dx} = 6x^2 - 2x + 6 \). This shows how each component of a polynomial takes part in forming the overall change rate.

Understanding these steps makes the process clear and showcases the power of differentiation in breaking down complex functions into manageable parts.