MATH 12002 - CALCULUS I §2.3 & §2.4: Derivatives of ... - Kent

[Pages:11]MATH 12002 - CALCULUS I ?2.3 & ?2.4: Derivatives of Trigonometric Functions

Professor Donald L. White

Department of Mathematical Sciences Kent State University

D.L. White (Kent State University)

1 / 11

Derivatives of Sine & Cosine

Our first two differentiation formulas for the trigonometric functions are

d

d

sin x = cos x and cos x = - sin x.

dx

dx

The first of these is proved in the text; we will prove the second using the definition of derivative. We will also need the angle sum formula for cosine,

cos(A + B) = cos A cos B - sin A sin B,

and the limits from Equation 6 and Example 11 of ?1.4,

sin

cos - 1

lim = 1 and lim

= 0.

0

0

D.L. White (Kent State University)

2 / 11

Derivatives of Sine & Cosine

By definition of the derivative, we have

d

cos(x + h) - cos x

cos x = lim

dx

h0

h

[cos x cos h - sin x sin h] - cos x

= lim

h0

h

cos x(cos h - 1) - sin x sin h

= lim

h0

h

cos h - 1

sin h

= lim cos x

- sin x

h0

h

h

cos h - 1

sin h

= lim cos x

- lim sin x

h0

h

h0

h

cos h - 1

sin h

= (cos x) lim

- (sin x) lim

h0

h

h0 h

= (cos x)(0) - (sin x)(1) = - sin x.

D.L. White (Kent State University)

3 / 11

Derivatives of Other Trigonometric Functions

Recall that the other trigonometric functions can be written in terms of sin x and cos x:

sin x tan x = ,

cos x

cos x cot x = ,

sin x

1 sec x = ,

cos x

1 csc x = .

sin x

We can use these relations and the derivatives of sin x and cos x to find the derivatives of all of the trigonometric functions.

D.L. White (Kent State University)

4 / 11

Derivatives of Other Trigonometric Functions

Using the Quotient Rule, we have

d

d sin x

tan x =

dx

dx cos x

=

d dx

(sin

x

)

cos x - sin x

d dx

(cos

x

)

cos2 x

(cos x) cos x - sin x(- sin x)

=

cos2 x

=

cos2 x + sin2 x cos2 x

=

1 cos2 x

= sec2 x.

Hence

d dx

tan x

=

sec2

x.

Similarly,

d dx

cot

x

=

- csc2

x.

D.L. White (Kent State University)

5 / 11

Derivatives of Other Trigonometric Functions

Again using the Quotient Rule, we have

d

d1

sec x =

dx

dx cos x

=

d dx

(1)

cos x - 1 ?

d dx

(cos

x

)

cos2 x

(0) cos x - 1(- sin x)

=

cos2 x

sin x

1 sin x

= cos2 x = cos x ? cos x = sec x tan x.

Hence

d dx

sec x

=

sec x

tan

x.

Similarly,

d dx

csc

x

=

- csc

x

cot

x.

D.L. White (Kent State University)

6 / 11

Derivatives of Other Trigonometric Functions

We now have

Derivatives of Trigonometric Functions

d dx

sin x

=

cos x

d dx

cos x

=

-

sin x

d dx

tan x

=

sec2

x

d dx

cot x

=

-

csc2

x

d dx

sec x

=

sec x

tan

x

d dx

csc

x

=

- csc x

cot x

Notes:

Derivatives involving cot x and csc x may show up in homework problems on WebAssign occasionally due to randomization, but they will not appear on any exams.

The derivative of sec x is the product, sec x tan x = (sec x)(tan x), and not the composite function, sec tan x = sec(tan x).

D.L. White (Kent State University)

7 / 11

Examples

3 sin x

1

Find

the

derivative

of

f

(x )

=

x

+

5x2 .

By the quotient rule

f (x) =

d dx

(3

sin

x

)

(x

+

5x 2)

-

(3

sin

x)

(x

+

5x

2)2

d dx

(x

+

5x 2)

=

(3

cos

x)( x

+

5x 2)

-

(3

sin

x

)(

1 2

x

-1/2

( x + 5x2)2

+

10x ) .

D.L. White (Kent State University)

8 / 11

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