Teach me calculus from absolute beginner cover topics such as functions , limits , differentiation, integration and differential equations explain in very detail but keep it fast paced Tell me how to differentiate each type of function algebric , trigonometric, inverse trigonometric etc and integration and also do some exercise questions

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Welcome to calculus! We'll start with functions, which are rules that assign exactly one output for each input. Functions are written as f of x equals y. There are different types of functions: algebraic functions like polynomials and rational functions, and transcendental functions like trigonometric and exponential functions. Here's a simple quadratic function f of x equals x squared. When we input a value like 1.5, the function gives us the output by squaring the input. Now let's explore limits, which describe how a function behaves as the input approaches a specific value. We write this as 'limit of f(x) as x approaches c equals L'. A key concept related to limits is continuity. A function is continuous at a point if it's defined there, the limit exists, and the function value equals the limit. Here's an example of a rational function with a removable discontinuity at x equals 1. When we factor the numerator, we can simplify to x plus 1, except at x equals 1 where the original function is undefined. The limit as x approaches 1 is 2, even though the function has a hole at that point. Now we move to differentiation, which is about finding the rate of change of a function. The derivative represents both the instantaneous rate of change and the slope of the tangent line at any point. We denote the derivative as f prime of x or dy/dx. Let's look at some key differentiation rules. The Power Rule states that the derivative of x to the power n equals n times x to the power n minus 1. For example, the derivative of x squared is 2x. Here we can see the function x squared in blue, and at the point x equals 1.5, the tangent line has a slope of 3, which is exactly what we get from the formula 2x. The red curve shows the derivative function 2x, which gives us the slope at any point on the original curve. Now let's explore integration, which is the reverse process of differentiation. There are two main types of integrals. Indefinite integrals find antiderivatives, written as the integral of f(x) dx equals F(x) plus C, where C is the constant of integration. Definite integrals calculate the area under a curve between two points, written as the integral from a to b of f(x) dx equals F(b) minus F(a). The Power Rule for integration states that the integral of x to the power n equals x to the power n plus 1 divided by n plus 1, plus C. For example, the integral of x squared is x cubed over 3 plus C. Here we're finding the area under the curve x squared from x equals 0.5 to x equals 2.5. Using the Fundamental Theorem of Calculus, we evaluate the antiderivative at the endpoints to get approximately 5.33 square units. Finally, let's look at differential equations, which relate a function with its derivatives. For example, the differential equation dy/dx equals 3x squared minus 1 can be solved by integration. Integrating both sides, we get y equals x cubed minus x plus C, where C is the constant of integration. The red curve shows the derivative function, and the colored curves show different solutions corresponding to different values of C. These are called particular solutions, while the general form with the arbitrary constant C represents a family of solutions. Calculus has numerous applications across disciplines. In physics, it's used to describe motion and forces. In engineering, it helps with optimization and signal processing. In economics, it models rates of change and helps maximize profit. And in computer science, it's fundamental to algorithms and machine learning.