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= Basics of Calculus =

'''Calculus''' is a branch of mathematics that studies how things change. It helps us understand motion, growth, and areas under curves. Calculus is divided mainly into two parts: '''Differential Calculus''' and '''Integral Calculus'''.

== Differential Calculus ==

Differential Calculus focuses on the concept of the '''derivative''', which represents the rate at which a quantity changes. For example, it tells us how fast a car is moving at any instant.

The derivative of a function <math>f(x)</math> with respect to <math>x</math> is denoted as:

<math>\frac{df}{dx} \text{ or } f'(x)</math>

The derivative is defined as the limit:

<math>
f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}
</math>

=== Example ===

If <math>f(x) = x^2</math>, then the derivative is:

<math>
f'(x) = \frac{d}{dx}(x^2) = 2x
</math>

This means at any point <math>x</math>, the slope of the curve <math>y = x^2</math> is <math>2x</math>.

== Integral Calculus ==

Integral Calculus deals with the '''integral''', which represents the accumulation of quantities, such as area under a curve.

The definite integral of a function <math>f(x)</math> from <math>a</math> to <math>b</math> is written as:

<math>
\int_a^b f(x) \, dx
</math>

It calculates the total area between the graph of <math>f(x)</math>, the x-axis, and the vertical lines <math>x=a</math> and <math>x=b</math>.

=== Example ===

Find the area under the curve <math>f(x) = x</math> from <math>x=0</math> to <math>x=3</math>:

<math>
\int_0^3 x \, dx = \left[ \frac{x^2}{2} \right]_0^3 = \frac{3^2}{2} - \frac{0^2}{2} = \frac{9}{2} = 4.5
</math>

== Summary ==

* '''Derivatives''' tell us how a function changes at any point (rate of change).
* '''Integrals''' tell us the total accumulation, like area under curves.
* Calculus is fundamental for physics, engineering, economics, and many sciences.

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''Calculus opens the door to understanding the world through mathematics!''