Taking derivatives

Contents
The big idea of differential calculus is the concept of the derivative, which essentially gives us the direction, or rate of change, of a function at any of its points. Learn all about derivatives and how to find them here.
51 exercises available

Derivative as slope of tangent line

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.

Derivative as instantaneous rate of change

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.

Derivative as a limit

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.

Formal definition of 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.

Differentiability

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).

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.

Review: Derivative basics

Review your conceptual understanding of derivatives with some challenge problems.

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.

Rational functions differentiation (intro)

Learn how the power rule can be used to differentiate some basic rational functions.

Radical functions differentiation (intro)

Learn how the power rule can be used to differentiate some basic radical functions.

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.

eˣ and ln(x) derivatives

The derivative of eˣ is eˣ. That's pretty amazing. The derivative of ln(x) is 1/x, which is just as surprising.

Review: Basic differentiation

Review your understanding of basic differentiation rules and your knowledge of the derivatives of common functions with some challenge problems.

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 proof

Go "behind the scenes" with Sal and learn how the chain rule is proved.

Quotient rule

The quotient rule says that the derivative of the quotient f(x)/g(x) is [f'(x)g(x)-f(x)g'(x)]/g²(x). This helps us find the derivative of a function which is a quotient of two other, more basic, functions.

Review: Product, quotient, & chain rule

Review your understanding of the product, quotient, and chain rules with some challenge problems.

Rational functions differentiation

Solve problems involving the derivatives of rational functions. For example, find f'(1) of f(x)=(x+2)/(x-3).

Solve problems involving the derivatives of radical functions. For example, find g'(5) of g(x)=∛(x²+5x-3).

Derivatives capstone

Review your function differentiation skills with some challenge problems.

Implicit differentiation introduction

Some two-variable relationships cannot be turned into a function, like the circle equation x²+y²=4. Implicit differentiation allows us to find the derivative of y with respect to x, even in such equations.

Implicit differentiation (advanced examples)

Tackle some more advanced problems of implicit differentiation. For example, find dy/dx for e^(xy²)=x-y.

Inverse trig functions differentiation

Implicit differentiation allows us to find the derivatives of arcsin(x), arccos(x), and arctan(x).

Derivatives of inverse functions

Implicit differentiation provides us with the relationship between the derivatives of inverse functions: if f and g are inverse functions, then f’(x)=1/(g’(f(x)). Get comfortable working with this relationship.

Disguised derivatives

Equipped with knowledge about the derivatives of all common functions, evaluate some limits that represent various derivatives.

Proofs for the derivatives of eˣ and ln(x)

The derivatives of eˣ and ln(x) may be simple, but proving them is a different story. Learn all about it here.

Logarithmic differentiation

In logarithmic differentiation, we find the derivative of the natural log of a function instead of the derivative of the function itself. It may surprise you, but this can sometimes actually be easier than regular differentiation, if not the only available option. For example, finding the derivative of xˣ.

Parametric & vector-valued function differentiation

Parametric and vector-valued functions basically take one input and return two outputs. How do we differentiate such functions? Find out here.

Review your knowledge of the advanced differentiation topics with some challenge problems.

Higher-order derivatives

The derivative is a function, and as such it has its own derivative! The same goes for the derivative of the derivative, and so forth. These are all called higher-order derivatives.

Higher-order derivatives (parametric & vector-valued functions)

Learn how to find the second derivatives (and any higher-order derivative) of parametric and vector-valued functions.

Polar curve differentiation

Find tangents to polar curves.