Problem 57 - Calculus Volume 1

Question

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=4 x^{2}$$

Step-by-Step Solution

Answer

The derivative of \(f(x) = 4x^2\) is \(f'(x) = 8x\).

Step 1: Understand the Definition of a Derivative

The derivative of a function, denoted as \( f'(x) \), is defined as the limit \( \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \). We will use this definition to find the derivative of the function \( f(x) = 4x^2 \).

Step 2: Substitute into the Formula

Substitute \( f(x) = 4x^2 \) into the definition of the derivative. Thus, we have \( \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{4(x + h)^2 - 4x^2}{h} \).

Step 3: Expand the Function

Expand \((x + h)^2\) which is \(x^2 + 2xh + h^2\). Substitute back to get \(\lim_{h \to 0} \frac{4(x^2 + 2xh + h^2) - 4x^2}{h}\).

Step 4: Simplify the Expression

Distribute the 4 to obtain \(\lim_{h \to 0} \frac{4x^2 + 8xh + 4h^2 - 4x^2}{h}\). Cancel \(4x^2\) with \(-4x^2\), resulting in \(\lim_{h \to 0} \frac{8xh + 4h^2}{h}\).

Step 5: Factor Out \(h\)

Factor an \(h\) from the numerator: \(\lim_{h \to 0} \frac{h(8x + 4h)}{h}\), which simplifies to \(\lim_{h \to 0} (8x + 4h)\).

Step 6: Evaluate the Limit

As \(h\) approaches 0, the term \(4h\) goes to 0. Thus, the limit becomes \(8x\).

Step 7: Conclusion

Therefore, the derivative of \(f(x) = 4x^2\) is \(f'(x) = 8x\).

Key Concepts

Definition of a Derivative

When you hear about derivatives in calculus, you might think of them as measures of how a function changes. They are fundamental to understanding how quantities vary with each other. The formal definition of a derivative at a point is given by the limit formula. This formula looks like \[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \].
This is a way to find the slope of the tangent line to the curve at any point on the function. It's essentially asking: how does the function change as you make a very small step from \(x\) to \(x+h\)?

• Difference Quotient: The expression \(\frac{f(x + h) - f(x)}{h}\) is known as the difference quotient.
• Limit as \(h\) Approaches Zero: The need to take the limit as \(h\) goes to zero is important because it gives us the instantaneous rate of change. Without this, we would only have an average rate of change over the interval \([x, x+h]\).

This concept underpins much of calculus and allows us to explore rates of change in various disciplines like physics, engineering, and economics.

Limit Process

The limit process is a crucial concept in calculus that helps in understanding the precise behavior of functions as we approach certain points. In the context of derivatives, it is about making \(h\), the difference between two points, as small as possible—essentially approaching zero.
To apply the limit process, follow these steps:

• Insert the function into the difference quotient: e.g., for \(f(x) = 4x^2\), fill in the formula \(\frac{f(x+h)-f(x)}{h}\).
• Manipulate the algebra: expand and simplify terms to make the function more manageable, as we did with \(4(x^2 + 2xh + h^2)\) compared to \(4x^2\).
• Once simplified, examine the behavior as \(h\) approaches 0. This step is pivotal, and anything that prevents this limit from being evaluated needs to be rectified, often by simplifying the expression further.

The process allows us to uncover the derivative, which is crucial to understanding instantaneous rates of change and the slope of curves.

Polynomial Differentiation

Polynomial functions are one of the simplest types of functions in mathematics, known for their smooth and continuous nature. When finding their derivatives, the process becomes quite straightforward once you grasp it, often involving some common rules.
To differentiate a polynomial function like \(f(x) = 4x^2\), observe that polynomials take the form of terms added together. Each term is a coefficient multiplied by a variable raised to a power. The differentiation rules for polynomials are:

• Power Rule: For \(x^n\), the derivative is \(nx^{n-1}\).
• Constant Multiplication Rule: If there is a constant multiplied by the function, you can differentiate inside and then multiply by the constant. So for \(4x^2\), you first differentiate \(x^2\) to obtain \(2x\), and multiply by 4 to get \(8x\).

These rules allow for quick differentiation of polynomials, and make calculating derivatives less tedious by reducing them to straightforward algebraic manipulations. In our example, this led us from \(f(x) = 4x^2\) to \(f'(x) = 8x\).