What is differential calculus and with some examples?

Differential calculus is a branch of calculus that deals with the study of rates of change of functions. It focuses on finding the derivative of a function, which represents the rate at which the function is changing at any given point.

Here are a few examples to illustrate the concept of differential calculus:

1. Example: Find the derivative of the function f(x) = 3x^2 + 2x - 1.
Solution: To find the derivative, we differentiate each term of the function separately. The derivative of 3x^2 is 6x (using the power rule), the derivative of 2x is 2 (using the constant rule), and the derivative of -1 is 0 (as the derivative of a constant is always 0). Therefore, the derivative of f(x) is f'(x) = 6x + 2.

2. Example: Find the derivative of the function g(x) = sin(x) + cos(x).
Solution: To find the derivative, we differentiate each term of the function separately. The derivative of sin(x) is cos(x) (using the derivative of sine rule), and the derivative of cos(x) is -sin(x) (using the derivative of cosine rule). Therefore, the derivative of g(x) is g'(x) = cos(x) - sin(x).

3. Example: Find the derivative of the function h(x) = e^x.
Solution: The derivative of e^x is simply e^x (using the derivative of exponential rule). Therefore, the derivative of h(x) is h'(x) = e^x.

These examples demonstrate how differential calculus allows us to find the rate of change of various functions, which is useful in many fields such as physics, economics, and engineering.