What is the difference between differential and integral calculus?

Differential Calculus vs. Integral Calculus

Differential Calculus is the branch of calculus that deals with rates of change and slopes of curves. Its primary focus is on finding derivatives, which represent the rate at which a quantity is changing at a specific point. Here are some key aspects of differential calculus:

• Fundamental Concept: The derivative, denoted as $f'(x)$ or $\frac{df}{dx}$, measures how a function changes at a specific point.
• Operations: It involves techniques like limit definitions, power rule, product rule, quotient rule, and chain rule to find derivatives.
• Applications: Differential calculus is used to find rates of change, maximum and minimum values of functions, and equations of tangents to curves.

Integral Calculus, on the other hand, is concerned with accumulating quantities and finding total values. It focuses on finding antiderivatives and definite integrals. Here's what you need to know about integral calculus:

• Fundamental Concept: The definite integral, denoted as $\int_a^b f(x) \, dx$, calculates the signed area between the curve $y = f(x)$ and the x-axis from $a$ to $b$.
• Operations: It involves techniques like the fundamental theorem of calculus, substitution, integration by parts, and partial fractions to find antiderivatives and definite integrals.
• Applications: Integral calculus is used to find total quantities, volumes, surface areas, and average values of functions.

In summary, while differential calculus focuses on rates of change and slopes, integral calculus is about accumulating quantities and finding total values. They are interconnected, with differentiation and integration being inverse operations.