Problem 109 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=8 x^{4}+9 x^{2}-1$$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 32x^3 + 18x\).

Step 1: Recognize the Function Structure

The function given is a polynomial function: \[ f(x) = 8x^4 + 9x^2 - 1 \] A polynomial function is a sum of terms, each consisting of a coefficient multiplied by a variable raised to an exponent.

Step 2: Apply the Power Rule to Each Term

To differentiate each term of the polynomial, use the power rule for differentiation. The power rule states that if \( f(x) = ax^n \), then the derivative \( f'(x) = anx^{n-1} \).

Step 3: Differentiate the First Term

Differentiate the first term \( 8x^4 \) using the power rule:\[ f'(x) = 4 imes 8x^{4-1} = 32x^3 \]

Step 4: Differentiate the Second Term

Differentiate the second term \( 9x^2 \) using the power rule:\[ f'(x) = 2 imes 9x^{2-1} = 18x \]

Step 5: Differentiate the Constant Term

Differentiate the constant term \(-1\). The derivative of a constant is zero:\[ f'(x) = 0 \]

Step 6: Combine the Derivatives

Combine all the derivatives obtained from steps 3, 4, and 5:\[ f'(x) = 32x^3 + 18x + 0 = 32x^3 + 18x \]The derivative of the function is thus \( f'(x) = 32x^3 + 18x \).

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression that consists of variables and coefficients, which are added and subtracted. Each term in a polynomial function is made up of a coefficient, a variable, and a non-negative integer exponent. For instance, in the polynomial function

• \( f(x) = 8x^4 + 9x^2 - 1 \),

- The terms are: \(8x^4\), \(9x^2\), and \(-1\).- Here, 8 and 9 are the coefficients, \(x^4\) and \(x^2\) are variable parts, and 4 and 2 are exponents.- The term \(-1\) is a constant because it doesn't have a variable part.Understanding polynomial functions is key to mastering calculus since they often appear in modeling real-world scenarios.

Power Rule

The power rule is a fundamental principle for finding the derivative of polynomial terms. It's a simple yet powerful tool that allows us to differentiate terms of the form \(ax^n\) with ease.- When you have a term like \(ax^n\), the power rule tells you to bring the exponent \(n\) down in front as a multiplier, and then decrease the exponent by one. This means:\[ \text{If } f(x) = ax^n, \text{ then } f'(x) = anx^{n-1}\]- By applying this rule, we can systematically determine the derivative of each term in a polynomial, simplifying complex differentiation into manageable steps.

First Derivative

The first derivative of a function is a way to find the rate of change or the slope of the function at any given point. In calculus, calculating the first derivative is crucial for analysing the behavior of functions.- For a function like \(f(x) = 8x^4 + 9x^2 - 1\), the process involves differentiating each term separately.- Using the power rule for the terms \(8x^4\) and \(9x^2\), and using the rule for constant differentiation (where the derivative of a constant is zero),- We find that \(f'(x) = 32x^3 + 18x\).This new function, \(f'(x)\), represents how \(f(x)\) moves and changes, making it a vital tool for sketching the shape of graphs and identifying key points of interest.

Constant Term Differentiation

In any polynomial, or mathematical expression for that matter, constants are terms without variables. When differentiating, constants follow a special straightforward rule: - The derivative of any constant is zero. Therefore, when differentiating a function like \(f(x) = 8x^4 + 9x^2 - 1\), despite the complexity of other terms, the constant term \(-1\) doesn't change. This rule ensures that differentiation doesn’t add unnecessary complexity.- If you encounter a constant term, remember:\[ \text{If } c \text{ is a constant, then its derivative is } 0\]This rule simplifies differentiation of expressions including constants, making it one less piece of computation to worry about.