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The first step in finding the first derivative is to understand the concept itself. A first derivative of a function gives us the rate at which the function is changing at any point. It is essentially the slope of the tangent line to the curve at any given point. For a given function \( y = f(x) \), the notation for the first derivative is usually \( y' \) or \( \frac{dy}{dx} \). The first derivative tells us whether the function is increasing or decreasing at each specific point.

In the given exercise, the function is \( y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4} \). Each term in the function needs to be differentiated independently using the power rule of differentiation, which simplifies the process.

Let's look closely at applying the power rule:

• For \( \frac{x^3}{3} \): Differentiate to get \( x^2 \).
• For \( \frac{x^2}{2} \): Differentiate to get \( x \).
• For \( \frac{x}{4} \): Differentiate to yield a constant \( \frac{1}{4} \).

Adding these results provides the first derivative: \( y' = x^2 + x + \frac{1}{4} \). This derivative expression gives a simplified way to understand how the original function is changing at any given \( x \) value.

The second derivative provides information about the curvature of the graph of the function, essentially the rate of change of the first derivative. It helps determine the concavity of a function: whether it is "concave up" (shaped like a cup \(\cup\)) or "concave down" (shaped like a cap \(\cap\)). The notation for the second derivative is \( y'' \) or \( \frac{d^2y}{dx^2} \).

In the example provided, the first derivative obtained is \( y' = x^2 + x + \frac{1}{4} \). This expression is then differentiated once more to find the second derivative. Using straightforward differentiation, we evaluate each term:

• For \( x^2 \): Differentiate to get \( 2x \).
• For \( x \): Differentiate to yield a constant \( 1 \).
• The constant \( \frac{1}{4} \) differentiates to \( 0 \).

Combining these results gives the second derivative: \( y'' = 2x + 1 \). This function indicates the concavity of the original function; if \( y'' > 0 \), the graph is concave up. If \( y'' < 0 \), the graph is concave down at that point.

The power rule of differentiation is one of the simplest and most commonly used rules in calculus, allowing for the quick differentiation of polynomial functions. The rule states that for any power function of the form \( x^n \), the derivative with respect to \( x \) is \( nx^{n-1} \). This means you multiply by the power, and then reduce the power by one.

The power rule is extremely useful when dealing with polynomial terms:

• When you have \( x^3 \), the derivative using the power rule is \( 3x^2 \).
• For \( x^2 \), the derivative becomes \( 2x \).
• Even a simple \( x \) is differentiated to \( 1 \), as \( x^{1-1} = x^0 = 1 \).

It’s important to understand that this rule applies to each term individually, making it straightforward to find derivatives for sums and differences of polynomial expressions. In our example, this rule enabled quick and accurate differentiation of each component of the function, leading to the first and second derivatives we discussed.