MATH 12002 - CALCULUS I 2.3 & 2.4: Derivatives of Trigonometric - - PowerPoint PPT Presentation

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 dx sin x = cos x and d dx cos x = − sin x. 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, lim

θ→0

sin θ θ = 1 and lim

θ→0

cos θ − 1 θ = 0.

D.L. White (Kent State University) 2 / 11

Derivatives of Sine & Cosine

By definition of the derivative, we have d dx cos x = lim

h→0

cos(x + h) − cos x h = lim

h→0

[cos x cos h − sin x sin h] − cos x h = lim

h→0

cos x(cos h − 1) − sin x sin h h = lim

h→0

• cos x cos h − 1

h − sin x sin h h

• =

lim

h→0

• cos x cos h − 1

h

• − lim

h→0

• sin x sin h

h

• =

(cos x) lim

h→0

cos h − 1 h

• − (sin x) lim

h→0

sin h 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: tan x = sin x cos x , cot x = cos x sin x , sec x = 1 cos x , csc x = 1 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 dx tan x = d dx sin x 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 dx sec x = d dx

• 1

cos x

• =

d

dx (1)

• cos x − 1 ·

d

dx (cos x)

• cos2 x

= (0) cos x − 1(− sin x) cos2 x = sin x cos2 x = 1 cos x · sin 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

1 Find the derivative of f (x) =

3 sin x √x + 5x2 . By the quotient rule f ′(x) = d

dx (3 sin x)

• (√x + 5x2) − (3 sin x)

d

dx (√x + 5x2)

• (√x + 5x2)2

= (3 cos x)(√x + 5x2) − (3 sin x)(1

2x−1/2 + 10x)

(√x + 5x2)2 .

D.L. White (Kent State University) 8 / 11

Examples

2 Find the derivative of F(x) = 3x5 tan x.

By the product rule, with f (x) = 3x5 and g(x) = tan x, F ′(x) = d dx (3x5)

• (tan x) + (3x5)

d dx (tan x)

• =

15x4 tan x + 3x5 sec2 x.

D.L. White (Kent State University) 9 / 11

Examples

3 Find the derivative of F(x) = x2 sin x cos x.

By the general product rule, with f (x) = x2, g(x) = sin x, and h(x) = cos x, F ′(x) = d

dx x2

sin x cos x + x2 d

dx sin x

• cos x + x2 sin x

d

dx cos x

• =

2x sin x cos x + x2 cos x cos x + x2 sin x(− sin x).

D.L. White (Kent State University) 10 / 11

Examples

4 Find the derivative of f (x) = sec x tan x

cos x . By the quotient rule f ′(x) = d

dx (sec x tan x)

• (cos x) − (sec x tan x)

d

dx (cos x)

• cos2 x

=

• (sec x tan x) tan x + sec x(sec2 x)
• cos x − (sec x tan x)(− sin x)

cos2 x .

D.L. White (Kent State University) 11 / 11