Problem 228 - Calculus Volume 1

Question

Find \(\frac{d y}{d x}\) for each function. $$ y=\left(3 x^{2}+3 x-1\right)^{4} $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 4(3x^2 + 3x - 1)^3 (6x + 3) \).

Step 1: Identify the Function Structure

The given function is \( y = (3x^2 + 3x - 1)^4 \). It is of the form \((u(x))^n\) where \(u(x) = 3x^2 + 3x - 1\) and \(n = 4\).

Step 2: Apply the Chain Rule

To find \( \frac{dy}{dx} \), apply the chain rule: \( \frac{dy}{dx} = n \cdot (u(x))^{n-1} \cdot \frac{du}{dx} \). Here, \( n = 4 \) and \( u(x) = 3x^2 + 3x - 1 \).

Step 3: Compute the Derivative \( \frac{du}{dx} \)

Calculate the derivative of \( u(x) = 3x^2 + 3x - 1 \). Use the power rule on each term:- \( \frac{d}{dx}[3x^2] = 6x \)- \( \frac{d}{dx}[3x] = 3 \)- \( \frac{d}{dx}[-1] = 0 \)Thus, \( \frac{du}{dx} = 6x + 3 \).

Step 4: Substitute and Simplify

Substitute \( n = 4 \), \( u(x) = 3x^2 + 3x - 1 \), and \( \frac{du}{dx} = 6x + 3 \) into the chain rule formula: \[ \frac{dy}{dx} = 4 \cdot (3x^2 + 3x - 1)^3 \cdot (6x + 3) \] Thus, the derivative \( \frac{dy}{dx} \) equals \( 4(3x^2 + 3x - 1)^3 (6x + 3) \).

Key Concepts

Chain Rule

The chain rule is a fundamental rule in calculus for finding the derivative of a composite function. A composite function is a function that is made of two or more functions. The chain rule essentially tells us how to differentiate these types of functions by breaking them into their constituent parts. To apply the chain rule, you should:

• Identify the outer function and the inner function. In the given problem, the outer function is ((u(x))^4 and the inner function is 3x^2 + 3x - 1.
• Find the derivative of the outer function. Keep the inner function as it is for now.
• Multiply the result by the derivative of the inner function.

Understanding the chain rule can greatly simplify differentiating complex functions, making it a vital tool in a mathematician's toolkit.

Power Rule

The power rule is another key concept in differentiation, providing a quick method for finding the derivative of functions of the form x^n, where is a constant exponent. The power rule formula is simple:

• If f(x) = x^n, then the derivative, f'(x), is nx^{n-1}.

Applying this rule breaks down into:

• For each term in the function, multiply the coefficient by the exponent.
• Subtract one from the exponent to find the new degree of the term.

In the provided exercise, the power rule was used to differentiate each term of the polynomial 3x^2 + 3x - 1. This is an essential rule in calculus, especially when dealing with polynomials.

Derivative of a Polynomial

Polynomials are algebraic expressions made up of terms consisting of variables raised to whole number powers and coefficients. The process of differentiating a polynomial involves applying the power rule to each term separately. In our example, the polynomial 3x^2 + 3x - 1 includes three terms:

• 3x^2, whose derivative is 6x,
• 3x, whose derivative is simply 3,
• and -1, which is a constant and its derivative is 0.

Finding the derivative of a polynomial is essentially about applying the power rule repeatedly. Once each term is differentiated, combine them to get the complete derivative. This method is straightforward and efficient, making it a fundamental skill when working with polynomials in calculus.