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When you hear the term "first derivative," think of it as the rate at which a function changes with respect to its variable, often denoted by \( x \). In simpler terms, it's like asking, "How fast is \( y \) changing as \( x \) changes?" For the exercise: \( y = -x^2 + 3 \).

- To find the first derivative, we differentiate \( y \) with respect to \( x \).- Differentiate using the power rule: for any term \( ax^n \), the derivative is \( anx^{n-1} \).
Applying the power rule here:- The term \( -x^2 \) becomes \( -2x \) after differentiation because you multiply the exponent by the coefficient and subtract one from the exponent.- The constant \( +3 \) has a derivative of \( 0 \) since constants do not change.

So, the first derivative, denoted as \( \frac{dy}{dx} \), is \( -2x \). This derivative allows us to understand how the function's output grows or shrinks as \( x \) moves.

The second derivative gives insight into how the rate of change itself is changing. It's a step further, observing the "acceleration" of the function if you will. If the first derivative tells you the speed, the second derivative tells you how that speed changes.

- For the exercise: The first derivative found was \( \frac{dy}{dx} = -2x \).- Now, differentiate \( -2x \) to find the second derivative.
Using the power rule:- The derivative of \( -2x \) is simply \( -2 \) because when you differentiate a term with an exponent of one, it reduces to the coefficient times a variable raised to the zero, which simplifies to the coefficient.

Thus, the second derivative, \( \frac{d^2y}{dx^2} \), is \( -2 \). This constant result tells us about the concavity of our function: it consistently bends downward.

One of the fundamental tools in calculus for finding derivatives is the power rule. The power rule is especially useful as it provides a quick method to determine the derivative of functions where the variable is raised to a power.

The power rule can be summarized as follows:

• Consider a function \( f(x) = ax^n \).
• The derivative, \( f'(x) \), is found using \( anx^{n-1} \).

Here's how it works in practice:- The exponent \( n \) is multiplied by the coefficient \( a \).- Then, the new exponent is the original reduced by one.

This powerful rule simplifies the process of differentiation, as showcased in our example:- For the term \( -x^2 \), the derivative is \( -2x \), adhering to \(-2x^{2-1}\).- Using the power rule, acknowledging constants disappear, simplifies calculating higher derivatives, like the second one for further change observation.