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The first derivative of a function measures how the function's output changes as its input changes. It's like seeing how the slope of a curve behaves at different points. To find the first derivative, we typically differentiate each term of the function separately.

For example, consider the function:

• \( y = \frac{4x^3}{3} - x \)

Using the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \), we differentiate each term:

• The derivative of \( \frac{4x^3}{3} \) is \( 4x^2 \). This is because we multiply the exponent (3) by the coefficient \( \frac{4}{3} \) and reduce the power by 1, yielding \( 4x^2 \).
• The derivative of \( -x \) is \( -1 \), as the derivative of \( x \) is simply 1.

Thus, the first derivative of the function is:

• \( \frac{dy}{dx} = 4x^2 - 1 \)

This expression gives us the slope of the function at any point \( x \). If you plug in a specific value of \( x \), you'll get the slope at that point.

The second derivative provides insight into the behavior of the function's curvature. Essentially, it tells us how the slope, given by the first derivative, changes as \( x \) changes. Is the slope getting steeper, or is it leveling out?

To find the second derivative, we differentiate the first derivative. Given the result from the first derivative:

• \( \frac{dy}{dx} = 4x^2 - 1 \)

We apply the power rule again to differentiate each term:

• The derivative of \( 4x^2 \) is \( 8x \). Here, we multiply the power (2) by the coefficient (4), yielding 8, and decrease the power by 1, resulting in \( 8x \).
• The derivative of the constant \( -1 \) is 0, as constants vanish when differentiated.

Thus, the second derivative is:

• \( \frac{d^2y}{dx^2} = 8x \)

This new function \( \frac{d^2y}{dx^2} \) describes the rate of change of the slope. If this value is positive, the function is concave up (like a bowl). If negative, it's concave down (like a frown).

The power rule is a fundamental tool in calculus that's very useful for finding derivatives of polynomial functions. It simplifies the process of differentiation significantly and can be described in one simple formula:

For any function \( y = x^n \), the derivative \( \frac{d}{dx} x^n \) equals to \( nx^{n-1} \).

Here's how it works:

• Identify the exponent \( n \) of the term \( x^n \).
• Multiply the coefficient of \( x^n \) by \( n \), the exponent.
• Decrease the power \( n \) by 1 to get \( n-1 \).

Applying the power rule lets you differentiate terms quickly without having to think about complex limits or other techniques. It turns a potentially complicated process into simple arithmetic.

For example, to differentiate \( 4x^3 \) using the power rule:

• The coefficient is 4, and \( n = 3 \).
• Multiply 4 by 3 to get 12, then reduce the power by 1, resulting in \( 12x^2 \).

This method is not only efficient but also reliable, forming a core part of any calculus student's toolkit.