# Taking derivatives

Contents

Calculating derivatives. Power rule. Product and quotient rules. Chain Rule. Implicit differentiation. Derivatives of common functions.

## Introduction to differential calculus

The topic that is now known as "calculus" was really called "the calculus of differentials" when first devised by Newton (and Leibniz) roughly four hundred years ago. To Newton, differentials were infinitely small "changes" in numbers that previous mathematics didn't know what to do with. Think this has no relevence to you? Well how would you figure out how fast something is going *right* at this moment (you'd have to figure out the very, very small change in distance over an infinitely small change in time)? This tutorial gives a gentle introduction to the world of Newton and Leibniz.

Newton, Leibniz, and Usain Bolt

Why we study differential calculus.

## Using secant line slopes to approximate tangent slope

The idea of slope is fairly straightforward-- (change in vertical) over (change in horizontal). But how do we measure this if the (change in horizontal) is zero (which would be the case when finding the slope of the tangent line. In this tutorial, we'll approximate this by finding the slopes of secant lines.

Slope of a line secant to a curve

What is the slope of a line between two points on a curve? This is called a "secant" line.

Slope of a secant line example 1

Computing the slope of a secant line of f(x) = ln(x)

Slope of a secant line example 2

An example finding the slope of a secant line based on numerical data for a function.

Slope of a secant line example 3

An example finding the slope of a secant line given points on a graph.

Instantaneous rate of change word problem

Building up to the idea of the derivative, this example asks us to estimate an instantaneous rate of change.

Approximating equation of tangent line word problem

Looking at numerical data for a function, we can estimate the equation for its tangent line.

Slope of secant lines

## Introduction to derivatives

Discover what magic we can derive when we take a derivative, which is the slope of the tangent line at any point on a curve.

Derivative as slope of a tangent line

Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line)

Tangent slope example 1

The slope of a tangent line can be expressed as the limit of certain secant lines.

Tangent slope example 2

Tangent slope example 3

Tangent slope

Test your understanding of a tangent line as the limit of secant slopes.

Slope of tangent line using derivative definition

The derivative can be thought of the slope of a tangent line, which is defined as the limit of the slopes of certain secant lines. See what this looks like in action.

The derivative of f(x)=x^2 for any x

Here we find a formula for the derivative of f(x)=x^2.

Tangent lines and rates of change

How tangent lines are a limit of secant lines, and where the derivative and rate of change fit into all this.

Formal and alternate form of the derivative

Compare different (equivalent) definitions for the derivative.

Formal and alternate form of the derivative for ln x

Enough abstract stuff already, let's see what the formal and alternative forms of the derivative look like in practice.

Formal and alternate form of the derivative example

A practice problem applying the different forms of the derivative.

The formal and alternate form of the derivative

Test your understanding of each definition of the derivative

## Visualizing graphs of functions and their derivatives

You understand that a derivative can be viewed as the slope of the tangent line at a point or the instantaneous rate of change of a function with respect to x. This tutorial will deepen your ability to visualize and conceptualize derivatives through videos and exercises.
We think you'll find this tutorial incredibly fun and satisfying (seriously).

Interpreting slope of a curve exercise

Recognizing slope of curves

Identify intervals with positive and negative slope on graphs of functions

Graphs of functions and their derivatives example 1

In this problem, we are about the value of a function and its derivative based on its graph.

Where a function is not differentiable

Find where a function is not differentiable.

Identifying a function's derivative example

Given the graph of a function, we are asked to recognize the graph of its derivative.

Figuring out which function is the derivative

More practice with the relationship between the graph of a function and the graph of its derivative.

Graphs of functions and their derivatives

After watching Sal go through some examples identifying what the graph of a derivative might look like, I bet you want to practice some yourself, don't you?

Intuitively drawing the derivative of a function

Rather than merely identifying the graph of the derivative of a function, a true test of understanding is to draw that derivative from a blank slate.

Intuitively drawing the antiderivative of a function

No we go the other way from what we did in the last video, drawing the graph of a function given what its derivative looks like.

Visualizing derivatives exercise

Sal walks through an exercise made by Stephanie Chang, where you fit a portion of the antiderivative of a function with the graph of the function itself.

Visualizing derivatives

Practice visualizing the derivative.

## Power rule

Calculus is about to seem strangely straight forward. You've spent some time using the definition of a derivative to find the slope at a point. In this tutorial, we'll derive and apply the derivative for any term in a polynomial.
By the end of this tutorial, you'll have the power to take the derivative of any polynomial like it's second nature!

Power rule

What is the formula for the derivative of a function f(x) = x^n

Is the power rule reasonable

Why does the power rule make sense?

Derivative properties and polynomial derivatives

Once you know how to take the derivative of x^n, it turns out you can take the derivative of any polynomial. Let's see why...

Power rule (basic)

Get started using the power rule for derivatives!

Power rule (advanced)

Want a bit more of a challenge applying the power rule?

Proof: d/dx(x^n)

Here Sal prooves the formula for the derivative of x^n using the binomial theorem.

Proof: d/dx(sqrt(x))

By request, Sal gives a proof of the derivative of the square root of x.

Using derivative properties

Given some values of the derivative of a function f, and the full definition of another function g, find the derivative of 3f(x)+2g(x)

Symbolic differentiation

Practice using the properties of the derivative.

## Derivatives of common functions

For your derivative-taking-toolbelt, it's good to know the derivative of several common functions such as sin(x), tan(x), e^x, etc.

Derivatives of sin x, cos x, tan x, e^x and ln x

Learn the derivatives of several common functions.

Special derivatives quiz

Test how well you know the derivatives of several common functions.

Special derivatives

Now that you know a few special derivatives, practice putting them into context.

## Chain rule

If you know the derivatives of f(x) and g(x), the chain rule gives you the power to find the derivative of f(g(x)) or g(f(x)). Believe me, this is no small power. Most complex and hairy functions can be thought of the composition of several simpler ones (ones that you can find derivatives of), so after this tutorial, you'll be able to take the derivative of almost any function.

Chain rule introduction

An introductory example of the chain rule for the function (sin(x))^2

Chain rule definition and example

See the general formula for the chain rule, as well as an example of what applying this formula looks like in practice.

Chain rule for derivative of 2^x

Even though the function 2^x doesn't look like a composition at first, the chain rule can help us find its derivative.

Derivative of log with arbitrary base

Using the chain rule, we can find the derivative of any logarithm based on the derivative of ln(x)

Chain rule example using visual function definitions

Chain rule example using visual information

Chain rule on two functions

Practice applying the chain rule.

Chain rule with triple composition

Why compose just two functions? Let's go crazy and compose three!

Derivative of triple composition

An example of the chain rule applied to a triple composition.

Chain rule on three functions

Solidify your understanding of the chain rule by practicing it on triple compositions.

Chain rule overview

A quick overview sheet of the chain rule.

Extreme derivative word problem (advanced)

A difficult but interesting derivative word problem

## Proving the chain rule

We've already been using the chain rule, but let's take a moment to really convince ourselves that it'll always work. It's not necessary to watch these videos before moving on, but if you have the time, they'll help to deepen your understanding of derivatives, continuity, and the chain rule.

Differentiability implies continuity

If a function has a derivative at all its points, it must be continuous. See why!

Change in continuous function approaches 0

Change in continuous function approaches 0

Chain rule proof

Here we use the formal properties of continuity and differentiability to see why the chain rule is true.

## Product and quotient rules

If you know the derivative of two functions f(x) and g(x), can you find the derivative of f(x)g(x)? What about f(x)/g(x)? Well, you this is exactly what you'll be able to do after this tutorial.

The product rule for derivatives

How to take the derivative of the product of two functions.

Product rule for more than two functions

How can you take the product of three functions? Four?

Product rule

Practice applying the product rule.

Quotient rule from product rule

Given the derivatives of f(x) and g(x), what is the derivative of f(x)/g(x)?

Quotient rule for derivative of tan x

Since tan(x) = sin(x)/cos(x), it is great candiate for applying the quotient rule.

Quotient rule

Practice applying the quotient rule.

Using the product rule and the chain rule

By combining the product rule and the chain rule, we can find the derivative of some truly complicated functions.

Combining the product and chain rules

Try your hand at taking the derivative of some hairy functions using the product rule together with the chain rule.

Product rule proof

Why does the product rule work?

## Implicit differentiation

Some curves, like the circle "x^2+y^2 = 1", are not defined as the graph of some function, as in "y = f(x)". After all, mathematical relations, like people, are not always explicit about their intentions. Here we learn how to find the slopes of these implicitly defined curves.

Implicit differentiation

Introduction to implicit differentiation.

Showing explicit and implicit differentiation give same result

Why does implicit differentiation do the same thing as ordinary differntiation?

Implicit derivative example 1

Implicit differentiation of of (x-y)^2 = x + y - 1

Implicit derivative example 2

Implicit differentiation of y = cos(5x - 3y)

Implicit derivative example 3

Implicit derivative of (x^2 +y^2) = 5(x^2)(2y^2)

Finding slope of tangent line with implicit differentiation

After practicing implicit diffentiation a few times, let's actually use it to find the slope of a tangent line.

Implicit derivative example 4

One more example of implicit differentiation.

Implicit differentiation

Practing using implicit differentiation.

## Derivatives of inverse functions

In this tutorial we explore a common method to find the derivatives of inverse tangent (arctangent), inverse sine (arcsine), inverse cosine (arccosine) and the natural logarithm function.

Derivative of inverse sine

Derivative of inverse sine

Derivative of inverse cosine

Derivative of inverse cosine.

Derivative of inverse tangent

More inverses! Inverse of tangent this time.

Derivative of natural logarithm

How can you find the derivative of ln(x) by viewing it as the inverse of e^x?

Derivatives of inverse functions

Derivatives of inverse functions including inverse trig

Derivative of x^(x^x)

This is actually a pretty fun derivative to be able to do.

Derivative using log properties

By exploiting our knowledge of logarithms, we can make certain derivatives much smoother to compute.

Logarithmic differentiation

You know your log properties, you know your derivatives, now practice using them both together.

## Proofs of derivatives of common functions

We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!

Proof: d/dx(ln x) = 1/x

Taking the derivative of ln x

Proof: d/dx(e^x) = e^x

Proof that the derivative of e^x is e^x.

Proofs of derivatives of ln(x) and e^x

Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".