# Derivative introduction

Contents

Get comfortable with the big idea of differential calculus, the derivative. The derivative of a function has many different interpretations and they are all very useful when dealing with differential calculus problems. This topic covers all of those interpretations, including the formal definition of the derivative and the notion of differentiable functions.

11 exercises available

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

One way of thinking about the derivative (of a function at a point) is as the slope of the tangent line (to the function's graph at that point). Get comfortable with this approach here.

One way of thinking about the derivative is as instantaneous rate of change. This is quite incredible because rate of change is usually found over a period of time, and not at an instant. Get comfortable with this approach here.

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

The derivative is actually a special kind of limit! This is yet another important stop on our continuing journey towards a formal definition of the derivative.

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.

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.

The derivative of a function isn't necessarily defined at every point. Learn about the conditions for the derivative to exist, and specifically about how continuity fits with this story (spoiler: for a function to be differentiable at a point it must be continuous at that point, but the other way isn't necessary).

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.

Review your conceptual understanding of derivatives with some challenge problems.