91Ó°ÊÓ

Calculus is the branch of mathematics that helps us study changes and motion. It focuses on concepts like limits, functions, derivatives, and integrals. The key idea is to analyze how a quantity changes as another quantity changes.

There are two main branches of calculus: differential calculus and integral calculus. Differential calculus is about finding the rate at which something changes. This rate is called a derivative. On the other hand, integral calculus is about accumulation of quantities and areas under curves.

In our exercise, we are using differential calculus to find the second derivative of a given function. This means we are finding out how the rate of change itself is changing.

Differentiation is the process of finding the derivative of a function. The derivative tells us the rate at which a function is changing at any point.

For example, if you have a function for the distance traveled over time, differentiating this function gives us the speed (rate of change of distance with respect to time). Differentiation can be applied multiple times. The first derivative gives us the rate of change, and the second derivative tells us how this rate is changing over time.

In the provided exercise, we first rewrote the given function using exponent notation:
\( y = \sqrt{x} \) was rewritten as \( y = x^{1/2} \)

Then we found the first derivative using the power rule, and used it to further differentiate and find the second derivative.

The power rule is a basic rule in differentiation that makes finding the derivative of polynomial functions straightforward. The rule states that if you have a function \( f(x) = x^n \), its derivative is \( f'(x) = nx^{n-1} \).

To use the power rule, follow these steps:

• Start with the exponent (n).
• Multiply the function by this exponent.
• Subtract one from the exponent to get the new exponent.

In the exercise, when we differentiated \( y = x^{1/2} \), we applied the power rule. The derivative of \( x^{1/2} \) is \( \frac{1}{2} x^{-1/2} \). When we found the second derivative, we applied the power rule again to differentiate \( \frac{1}{2} x^{-1/2} \). The process of using this rule is essential in solving calculus problems efficiently.