Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=\frac{1}{3}\left(2 x^{3}-4\right) \quad\left(0,-\frac{4}{3}\right) $$

Step-by-Step Solution

Answer

The value of the derivative of the function at the given point (0, -4/3) is 0. The power rule was used for differentiation.

Step 1: Identify the Function and Rule

Identify the function, which is \(f(x) = \frac{1}{3}(2x^3 - 4)\). Our polynomial function can be differentiated using the power rule. The power rule states that the derivative of \(x^n\), where \(n\) is any real number is \(nx^{n−1}\).

Step 2: Apply the Power Rule

Apply the power rule on each term of the function. Derivative of \(2x^3\) is \(3*2x^{3-1} = 6x^2\), and derivative of constant term -4 is 0. Since we multiplied 1/3 with each term at the start, our derivative function becomes \(f'(x) = \frac{1}{3}(6x^2) = 2x^2\).

Step 3: Substitute the Given Point

Substitute the x-coordinate from the point (0, -4/3) into the derivative function. This gives us \(f'(0) = 2*(0)^2 = 0\).

Key Concepts

Differentiation Rules

Differentiation is a fundamental concept in calculus. It deals with finding how a function changes as its inputs change. Differentiation rules help simplify this process, especially when dealing with complex functions. The main differentiation rules include:

• Power Rule: Used for functions of the form \(x^n\), where the derivative is \(nx^{n-1}\).
• Product Rule: Useful for functions that are products of two simpler functions. If \(u\) and \(v\) are functions, then their derivative is \(u'v + uv'\).
• Quotient Rule: Applied to functions that are ratios of two functions. For functions \(u/v\), the derivative is \((u'v - uv')/v^2\).
• Chain Rule: Used when dealing with composite functions. If you have a function \(g(f(x))\), the derivative is \(g'(f(x))f'(x)\).

Each of these rules helps tackle specific forms of functions effectively, making the process of finding derivatives a lot faster and more systematic.

Polynomial Differentiation

Polynomials are one of the simplest kinds of mathematical functions. They consist of expressions like \(ax^n + bx^{n-1} + ... + c\). Differentiating polynomials is straightforward when we use the power rule.
Polynomial differentiation involves taking the derivative of each term separately. Here are steps to follow when differentiating a polynomial:

• Identify each term in the polynomial.
• Apply the power rule to each term accordingly.
• Combine the results to obtain the derivative of the entire polynomial.

For example, when differentiating \(2x^3 - 4\), we apply the power rule to \(2x^3\) which gives \(6x^2\), and the constant \(-4\) gives \(0\). Thus, the derivative of the polynomial is simply \(6x^2\). This step-by-step approach makes polynomial differentiation predictable.

Power Rule

The power rule is one of the most essential differentiation rules. It's particularly convenient, as it reduces the complexity of finding derivatives of polynomials and other power functions.
According to the power rule, if you have a term \(x^n\), its derivative is \(nx^{n-1}\). This means you take the exponent, multiply it by the coefficient, and then subtract one from the exponent.
For example, consider the function \(f(x) = 2x^3\):

• The exponent \(n\) is \(3\).
• Multiply \(3\) by the coefficient \(2\), giving you \(6\).
• Subtract one from the exponent \(3\), resulting in \(2\).
• The derivative is \(6x^2\).

Such a method makes it direct and efficient to find the slope of the tangent line at any point on a curve described by a polynomial, greatly simplifying the task of finding how functions change.