Problem 109 - Calculus Volume 1

Question

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: Identify the Problem

The problem asks us to find the derivative of the polynomial function \(f(x) = 8x^4 + 9x^2 - 1\). This is represented as \(f'(x)\).

Step 2: Apply the Power Rule

The power rule for differentiation states that if \(f(x) = ax^n\), then \(f'(x) = nax^{n-1}\). We will apply this rule to each term in the polynomial separately. First, consider \(8x^4\) which becomes \(4 \times 8x^{4-1} = 32x^3\).

Step 3: Differentiate the Second Term

For the second term \(9x^2\), apply the power rule: \(2 \times 9x^{2-1} = 18x^1 = 18x\).

Step 4: Differentiate the Constant

The derivative of a constant is zero. Therefore, the derivative of \(-1\) is \(0\).

Step 5: Combine the Results

Combine the derivatives of each term to get the complete derivative of the function: \(f'(x) = 32x^3 + 18x\).

Step 6: Final Step: State the Derivative

Thus, the derivative of \(f(x) = 8x^4 + 9x^2 - 1\) is \(f'(x) = 32x^3 + 18x\).

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus that simplifies how we find derivatives of polynomial terms. It's really all about reducing complex expressions into something more straightforward.
For any term of the form \(ax^n\), the power rule tells us how to find its derivative: simply multiply the exponent \(n\) by the coefficient \(a\), and then subtract one from the exponent. This gives us the derivative formula: \(f'(x) = nax^{n-1}\).
Let's take a quick example: if you have \(4x^3\), applying the power rule involves multiplying the 3 by 4 (making it 12) and subtracting 1 from the power of \(x\). So, the derivative is \(12x^2\).
This rule is intuitive once you see it applied. It's like peeling away a layer from the power of the polynomial, making it simpler each time you differentiate.

Differentiation

Differentiation is the process of finding a derivative. Essentially, it measures how a function changes as its input changes. In our example function \(f(x) = 8x^4 + 9x^2 - 1\), differentiation helps us find the rate of change of the function at any point \(x\).
What's amazing about differentiation is that it allows us to break down complex functions into simpler, more manageable pieces. This splitting makes it easier to understand how a function behaves.

• For the term \(8x^4\), differentiation simplifies it to \(32x^3\).
• The term \(9x^2\) differentiates to \(18x\).
• Constant terms like \(-1\) are reduced to zero, as their rate of change is non-existent.

Once you differentiate each term, you combine them back together, giving you the derivative of the whole polynomial, \(f'(x) = 32x^3 + 18x\). This new function, \(f'(x)\), shows the slope or rate of change of the original function's graph at any given \(x\).

Polynomial Functions

Polynomial functions are expressions that involve powers of \(x\) with coefficients. They are fundamental building blocks in mathematics due to their varied and easily understandable structure. In the function \(f(x) = 8x^4 + 9x^2 - 1\), each term is a power of \(x\) with a constant coefficient.
Polynomials are versatile: they can represent a wide array of different graphs and behaviors. By understanding these functions, you can predict how they act under transformations like differentiation.
The beauty of polynomial functions lies in their predictability when applying calculus principles. With each term being a simple product of constants and variables raised to integer powers, differentiation becomes straightforward. You apply the rules term by term, analyzing each contribution to the whole function individually.
This makes polynomial functions excellent candidates for practicing differentiation, especially with methods like the power rule. Whether it's a cubic function or a quintic one, the clarity and completeness of outcomes when differentiated using the power rule are what make polynomials so great for learning calculus fundamentals.