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Unit: Differentiation for physics (Prerequisite)

Slope

Learn about slope (the steepness of a line) and how to find it.

Introduction to differential calculus

What's differential calculus all about? An answer to this question lies just right here.

Secant lines

Gain some experience working with secant lines. This will help us on our journey to find a formal definition for the derivative.

Limit basics

A limit is the value that a function or sequence "approaches" as the input or index approaches some value. In this tutorial, we supply an intuitive understanding of limits.

Derivative as a limit

There are two ways to define the derivative of function f at point x=a. The formal definition is the limit of [f(a+h)-f(x)]/h as h approaches 0, and the alternative definition is the limit of [f(x)-f(a)]/(x-a) as x approaches a. Make introduction with these two definitions.

Using the formal definition of derivative

Learn how we can use their formal definition in order to find the derivatives of specific functions. For example, we find the derivative of f(x)=x² at x=3, or for any x-value.

Derivative as a function

This may blow your mind, but the derivative of a function is a function in itself! Get comfortable in thinking about the derivative as a function that is separate from, but tightly related to, its original function.

Basic differentiation rules

If you ever tried to find derivatives using their formal definition, you probably know how tedious that may be. Fortunately, we have ways for finding derivatives much quicker, using differentiation rules! Make your first steps in this fascinating world by working with the more basic rules. For example, the derivative of [f(x)+g(x)] is f'(x)+g'(x), and the derivative of k⋅f(x) is k⋅f'(x).

Power rule

The power rule says that the derivative of xⁿ is n⋅xⁿ⁻¹. It allows us to quickly find the derivative of any polynomial, and it doesn't even stop there! Make introduction with this simple but powerful rule.

Polynomial functions differentiation

Put the power rule to use by differentiating various polynomials.

Sine & cosine derivatives

The derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). How convenient! Practice differentiating functions that include sine and cosine.

Product rule

The product rule says that the derivative of the product f(x)g(x) is f'(x)g(x)+f(x)g'(x). This helps us find the derivative of a function which is a product of two other, more basic, functions.

Chain rule

The chain rule says that the derivative of the composite function f(g(x)) is f'(g(x))⋅g'(x). This helps us find the derivative of a composite function. It may be slightly hard to grasp, but its importance cannot be overstated!