Derivatives

DerivativesElaine Jasmine Darren Mike

Derivative of Trigonometric Functions

Introduction to the DerivativeDerivative of the Function

Differentiability & Continuity

Derivative PropertiesThe Constant Rule

The Power Rule

The Constant Multiple Rule

The Sum and Difference Rules

The Chain Rule

The Product Rule

The Quotient Rule

Higher Order Derivatives

Content

Problems

Introduction to the Derivative

Derivative of the Function

Tangent Line: The line only intersects a curve at one point.

Secant Line:The line intersects a curve at two points.

To understand what is the derivative of the function, we have to know these definitions first:

Slope Formula: m =

The Derivative at a Point

The derivative at a point is the slope of the tangent line at this point.

x -x2 1

y -y 2 1

0 x

y

2x1x

2y

1y

1

x

y

0 1a

f(a )1

x

y

1a

f(a +h)

1f(a )

1a +h0

limh 0

f(a+h)-f(a)hf’(a)=

The Derivative of the Function The derivative function tells us the value of the derivative for any

point on the original function. [1]

f’(x), ddx

, y’, [f(x)]ddx

, Dx[y].

Notations of the Derivative

limh 0

f(x+h)-f(x)hf’(x)=

[1] Thomas S. Downey. ”Derivative function.” http://www.calculusapplets.com/derivfunc.html.

Differentiability & Continuity• A function is differentiable on an interval contains point c if the

one-side limits and exists and are equal.

• A function has no derivative if

limx c+

f(x)-f(c)x-c

limx c -

f(x)-f(c)x-c

x

y

0 x

y

0 x

y

0 c

a. f(c) is discontinuous.

Removable Discontinuity

Jump Discontinuity

Infinite Discontinuity

Differentiability Implies Continuity

If f is differentiable at x=c, then f is continuous at x=c.

x

y

0

x

y

0 x

y

0

b. Vertical tangent line exists.

c. Sharp corners exist.

You will get two slopes at these red points.

Derivative Properties

The Constant Rule

ddx [c] = 0

x

y

0

C

The derivative of a constant is zero; that is, for a constant c: [2]

[2] Anonymous. “Basic Derivative Rules.” http://www.math.brown.edu/UTRA/derivrules.html. August 6, 2008.

The Power Rulen-1f’(x) = nax

ddx [x] =1

Before proving The Power Rule, we should introduce The Binomial Theorem:

[4] Donna Roberts. “The Binomial Theorem.” http://www.regentsprep.org/Regents/math/algtrig/ATP4/bintheorem.htm

[4]

• Proof: We can prove the power rule by using the Binomial Theorem. Consider the following proof of the power rule. [3]

If , then

Now, to find the derivative, we need to expand (x+h) using the Binomial Theorem. Doing so, we get the following equivalent statements: [3]

[3] Erin Horst. “Rules of Differentiation: The Power Rule.” http://jwilson.coe.uga.edu/EMAT6680/Horst/derivativepower/derivativepower.html

• Graph:

y=x 3

y’=3x 2

The following graph illustrates the function and its derivative

[3]

[3] Erin Horst. “Rules of Differentiation: The Power Rule.” http://jwilson.coe.uga.edu/EMAT6680/Horst/derivativepower/derivativepower.html

The Constant Multiple Rule

ddx [cf(x)] = cf’(x)

[cx ] = cnx n n-1ddx

The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the original function: [2]

[2] Anonymous. “Basic Derivative Rules.” http://www.math.brown.edu/UTRA/derivrules.html. August 6, 2008.

Plug in The Power Rule:

• Proof:

The Sum & Difference Rulesddx [f(x)+g(x)] = f’(x)+g’(x)

• Proof:

[2] Anonymous. “Basic Derivative Rules.” http://www.math.brown.edu/UTRA/derivrules.html. August 6, 2008.

[2]

The Chain RuleIf y=f(g(x)), then y’=f’(g(x)) g’(x)

The Product Ruleddx [f(x)g(x)] = f(x)g’(x)+f’(x)g(x)

The Quotient Ruled

dx [ ]f(x)g(x) = g(x)f’(x)-f(x)g’(x)

[g(x)]2 , g(x) ≠ 0

[2] Anonymous. “Basic Derivative Rules.” http://www.math.brown.edu/UTRA/derivrules.html. August 6, 2008.

• Proof: [2]

[2] Anonymous. “Basic Derivative Rules.” http://www.math.brown.edu/UTRA/derivrules.html. August 6, 2008.

• Proof: [2]

Derivative of Trigonometric

Functions

ddx[sinx] = cosx

ddx [cosx] = -sinx

ddx[tanx] = sec x2

ddx [secx] = secxtanx

ddx [cscx] = -cscxcotx

ddx

2[cotx] = -csc x

• Graph:

The following graph illustrates the function y=2tan(x) and

its derivative y’=2sec (x)2

[3] Erin Horst. “Derivative of Trigonometric Functions.” http://jwilson.coe.uga.edu/EMAT6680/Horst/derivativepower/derivativepower.html

Higher Order Derivatives

When more than one derivative is applied to a function, it is consider higher order derivatives.

For example, the derivative of y = x7 is 7x6. 7x6 is differentiable and its derivative is 42x5. 42x5 is called the second order derivative of y with respect to x for function x7. Repeating this process, function y's third order derivative 210x4 is obtained. [5]

[5] Hengzhong Wen, Chean Chin Ngo, Meirong Huang, Kurt Gramoll. “Higher Order Derivatives.” https://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=ma&chap_sec=03.1&page=theory

Work Cited[1] Thomas S. Downey. ”Derivative function.” http://www.calculusapplets.com/

derivfunc.html.

[2] Anonymous. “Basic Derivative Rules.” http://www.math.brown.edu/UTRA/derivrules.html. August 6, 2008.

[3] Erin Horst. “Rules of Differentiation: The Power Rule.” http://jwilson.coe.uga.edu/EMAT6680/Horst/derivativepower/derivativepower.html

[4] Donna Roberts. “The Binomial Theorem.” http://www.regentsprep.org/Regents/math/algtrig/ATP4/bintheorem.htm

[5] Hengzhong Wen, Chean Chin Ngo, Meirong Huang, Kurt Gramoll. “Higher Order Derivatives.” https://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=ma&chap_sec=03.1&page=theory

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