18/2/2014 Taking derivatives | Calculus | Khan Academy

https://www.khanacademy.org/math/calculus/differential-calculus 1/4

CALCULUS

Community Questions

Limits

Taking derivatives

Derivative applications

Indefinite and definite integrals

Solid of revolution

Sequences, series and function

approximation

AP Calculus practice questions

Double and triple integrals

Partial derivatives, gradient,

divergence, curl

Line integrals and Green's theorem

Surface integrals and Stokes'

theorem

Divergence theorem

Taking derivatives

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.

Using secant line slopes toapproximate 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.

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.

SUBSCRIBE

Newton, Leibniz, and Usain Bolt

Slope of a line secant to a curve

Slope of a secant line example 1

Slope of a secant line example 2

Slope of a secant line example 3

Approximating instantaneous rate of change word problem

Approximating equation of tangent line word problem

Slope of secant lines

Derivative as slope of a tangent line

Tangent slope as limiting value of secant slope example 1

Tangent slope as limiting value of secant slope example 2

Tangent slope as limiting value of secant slope example 3

Tangent slope is limiting value of secant slope

Calculating slope of tangent line using derivative definition

Derivatives 1

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

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Visualizing graphs of functions andtheir 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).

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!

Formal and alternate form of the derivative

Formal and alternate form of the derivative for ln x

Formal and alternate form of the derivative example 1

The formal and alternate form of the derivative

Interpreting slope of a curve exercise

Recognizing slope of curves

Calculus: Derivatives 1

Calculus: Derivatives 2

Derivative intuition module

Derivative intuition

Graphs of functions and their derivatives example 1

Where a function is not differentiable

Identifying a function's derivative example

Figuring out which function is the derivative

Graphs of functions and their derivatives

Intuitively drawing the derivative of a function

Intuitively drawing the antiderivative of a function

Visualizing derivatives exercise

Visualizing derivatives

Power rule

Is the power rule reasonable

Derivative properties and polynomial derivatives

Power rule

Proof: d/dx(x^n)

Proof: d/dx(sqrt(x))

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Chain rule

You can take the derivatives of f(x) and g(x), but

what about f(g(x)) or g(f(x))? The chain rule gives us

this ability. Because most complex and hairy

functions can be thought of the composition of

several simpler ones (ones that you can find

derivatives of), you'll be able to take the derivative

of almost any function after this tutorial. Just

imagine.

Product and quotient rules

You can figure out the derivative of f(x). You're

also good for g(x). But what about f(x) times g(x)?

This is what the product rule is all about. This

tutorial is all about the product rule. It also covers

the quotient rule (which really is just a special case

of the product rule).

Power rule introduction

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

Special derivatives

Chain rule introduction

Chain rule definition and example

Chain rule for derivative of 2^x

Derivative of log with arbitrary base

Chain rule example using visual function definitions

Chain rule example using visual information

Chain rule on two functions

Chain rule with triple composition

Derivative of triple composition

Chain rule on three functions

Extreme derivative word problem (advanced)

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

Special derivatives

Applying the product rule for derivatives

Product rule for more than two functions

Product rule

Quotient rule from product rule

Quotient rule for derivative of tan x

Quotient rule

Using the product rule and the chain rule

Product rule

Quotient rule and common derivatives

Equation of a tangent line

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Implicit differentiation

Like people, mathematical relations are not

always explicit about their intentions. In this

tutorial, we'll be able to take the derivative of one

variable with respect to another even when they

are implicitly defined (like "x^2 + y^2 = 1").

Proofs of derivatives of commonfunctions

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!

Equation of a tangent line

Implicit differentiation

Showing explicit and implicit differentiation give same result

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

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

Implicit derivative of (x^2+y^2)^3 = 5x^2y^2

Finding slope of tangent line with implicit differentiation

Implicit derivative of e^(xy^2) = x - y

Derivative of x^(x^x)

Implicit differentiation

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

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

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

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