91Ó°ÊÓ

Calculus is the branch of mathematics that studies continuous change. It is divided into two main parts: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. In this exercise, we are dealing with differential calculus, where we find the rates at which functions change. By understanding how these rates work, we can solve problems related to motion, growth, and many more real-world scenarios.

Differentiation is the process of finding the derivative of a function. The derivative shows how a function changes as its input changes. For instance, if we have a function representing the position of an object over time, its derivative would tell us the object's velocity. The first step in the given exercise involves finding the first derivative of the function. We start with the function \( y = 4x^2 - 5x + 7 \) and use differentiation rules to find how it changes with respect to \( x \). After calculating the first derivative, \( dy/dx = 8x - 5 \), we move forward to find the second derivative. This shows the rate at which the rate of change is changing, giving us deeper insights.

The power rule is a basic and essential rule in differentiation. It states that if you have a term \( x^n \), its derivative is \( nx^{n-1} \). In our exercise, we use the power rule to differentiate each term of the function. For the term \( 4x^2 \), we calculated \( d/dx [4x^2] = 8x \). Similarly, for the term \( -5x \), we found \( d/dx [-5x] = -5 \). Constant terms, like 7, have derivatives of 0 since they do not change as \( x \) changes. By applying the power rule, we simplify the process of finding both the first and the second derivatives. The second derivative, \( d^2y/dx^2 = 8 \), reflects the constant rate of change of the slope of the original function.