Chapter 2: Differentiation
Section 2.7: Derivatives of the Trig Functions
Essentials
Introduction
Table 2.7.1 lists the derivatives of the six trigonometric functions.
Function
Maple's Derivative
Textbook Derivative
sine
ⅆⅆ x sinx = cosx
cosx
cosine
ⅆⅆ x cosx = −sinx
−sinx
tangent
ⅆⅆ x tanx = 1+tanx2
sec2x
cotangent
ⅆⅆ x cotx = −1−cotx2
−csc2x
secant
ⅆⅆ x secx = secxtanx
secxtanx
cosecant
ⅆⅆ x cscx = −cscxcotx
−cscxcotx
Table 2.7.1 Derivatives of the six trigonometric functions
Maple returns the derivatives of the tangent and cotangent functions in a form that differs from the form found in the typical calculus textbook. (Note the red cells in Table 2.7.1.) This is because early-on, the Maple programmers opined that for these two functions, returning the derivative in terms of the same function was somehow "simpler" than returning it in terms of a different function. Table 2.7.2 contains Maple code for modifying the differentiation rules for these two functions. (The restart is in deference to any remember-table issues.)
restart; unprotect(`diff/tan`): unprotect(`diff(cot`): `diff/tan` := proc(a,x) sec(a)^2*diff(a,x);end: `diff/cot` := proc(a,x) -csc(a)^2*diff(a,x);end:
Table 2.7.2 Code to modify the differentiation rules for tangent and cotangent
If the differentiation rules are arranged as in Table 2.7.3, certain relations between all the trig functions and their derivatives can be observed.
ddx sinx=cosx
ddx cosx= −sinx
ddx tanx=sec2x
ddx cotx=−csc2x
ddx secx=secxtanx
ddx cscx= −cscxcotx
Table 2.7.3 Derivatives of the trigonometric functions
If each function in the left-hand column is replaced by its co-function, and a minus sign inserted for the derivative, the right-hand column results.
The derivatives of the sine and cosine functions follow from an application of Definition 2.2.1, the definition of the derivative, and from the two special limits in Table 1.4.1. The derivatives of the remaining functions are obtained by applying the Quotient, Power, and Chain rules to the forms of the functions shown in Table 2.7.4.
tanx=sinxcosx
cotx=cosxsinx
secx=1cosx
cscx=1sinx
Table 2.7.4 Expressing trig functions in terms of sine and cosine
Derivative of the Sine Function
A graph of the sine function suggests that this function is continuous and has a tangent line at all points. This suggests that the sine function should be differentiable everywhere.
Figure 2.7.1 shows the result of applying the to the sine function.
Figure 2.7.1 Application of the Derivative Tutor to the function sinx
The red curve is the graph of sinx, whereas the blue curve is the graph of its derivative. This blue curve seems to be periodic with period 2 π, and amplitude 1. Moreover, it seems to be a translation of the red curve, that is, a translation by π/2 of sinx. This suggests that the derivative of the sine function might indeed be the cosine function. In Table 2.7.5, an application of Definition 2.2.1 verifies that this conjecture is correct.
ddx sinx
=limh→0sin(x+h)−sin(x)h
=limh→0sin(x)cos(h)+cos(x)sin(h)−sin(x)h
=limh→0(sin(x)(cos(h)−1)h+cos(x)sin(h)h)
=sin(x)limh→0cos(h)−1h+cos(x)limh→0sin(h)h
=sin(x)0+cos(x)1
=cosx
Table 2.7.5 Application of Definition 2.2.1 to the sine function
Table 2.7.6 lists the higher derivatives of the sine function.
<<k,seq(k,k=0..8)>|<d/dx^k,seq(diff(sin(x),[x$k]),k=0..8)>>;
Table 2.7.6 Higher derivatives of sinx
The convention is that ddx0 fx = f x, that is, the zeroth-ordered derivative is the function itself since no derivative is taken. The table implies that the higher-order derivatives of the sine function are cyclic, with period 4.
Derivative of the Cosine Function
A graph of the cosine function suggests that this function is continuous and has a tangent line at all points. This suggests that the cosine function should be differentiable everywhere.
Figure 2.7.2 shows the result of applying the to the cosine function.
Figure 2.7.2 Application of the Derivative Tutor to the function cosx
The red curve is the graph of cosx, whereas the blue curve is the graph of its derivative. This blue curve seems to be periodic with period 2 π, and amplitude 1. Moreover, it seems to be a translation of the red curve, that is, a translation by π/2 of cosx. This suggests that the derivative of the cosine function might indeed be the sine function. In Table 2.7.7, an application of Definition 2.2.1 verifies that this conjecture is correct.
ddx cosx
=limh→0cos(x+h)−cos(x)h
=limh→0cos(x)cos(h)−sin(x)sin(h)−cos(x)h
=limh→0(cos(x)(cos(h)−1)h−sin(x)sin(h)h)
=cos(x)limh→0cos(h)−1h−sin(x)limh→0sin(h)h
=cos(x)0−sin(x)1
=−sin(x)
Table 2.7.7 Application of Definition 2.2.1 to the cosine function
Table 2.7.8 lists the higher derivatives of the sine function.
<<k,seq(k,k=0..8)>|<d/dx^k,seq(diff(cos(x),[x$k]),k=0..8)>>;
Table 2.7.8 Higher derivatives of sinx
The convention is that ddx0 fx = f x, that is, the zeroth-ordered derivative is the function itself since no derivative is taken. The table implies that the higher-order derivatives of the cosine function are cyclic, with period 4.
Derivative of the Tangent Function
Table 2.7.9 contains the calculations by which the derivative of the tangent function is obtained. The differentiation begins with an application of the Quotient rule.
ddx tanx
=ⅆⅆxsin(x)cos(x)
=cosxⅆⅆxsinx−sinxⅆⅆxcosxcos2x
=cosxcosx+sinxsinxcos2x
=cos2x+sin2xcos2x
=1cos2x
=sec2x
Table 2.7.9 Derivative of the tangent function
Derivative of the Cotangent Function
Table 2.7.10 contains the calculations by which the derivative of the cotangent function is obtained. The differentiation again begins with an application of the Quotient rule.
ddx cotx
=ddx cosxsinx
=sinx ddx cosx−cosx ddx sinxsin2x
=sinx−sinx−cosx cosxsin2x
= −sin2x+cos2xsin2x
= −1sin2x
= −csc2x
Table 2.7.10 The derivative of the cotangent function
Derivative of the Secant Function
Table 2.7.11 contains the calculations by which the derivative of the secant function is obtained. The differentiation begins with an application of the Power rule, followed by an application of the Chain rule.
ddx secx
=ddx 1cosx
=ⅆⅆxcos(x)−1
= −1 cosx−2ddx cosx
= −1 cosx−2sinx
=sinxcos2x
=1cos(x) sin(x)cos(x)
=sec(x)tan(x)
Table 2.7.11 Derivative of the secant function
Derivative of the Cosecant Function
Table 2.7.12 contains the calculations by which the derivative of the cosecant function is obtained. The differentiation again begins with an application of the Power rule, followed by an application of the Chain rule.
ddx cscx
=ddx 1sinx
=ddx sinx−1
= −1 sinx−2 ddx sinx
= −sinx−2 cosx
= −cosxsin2x
= −1sinx cosxsinx
= −cscx cotx
Table 2.7.12 Derivative of the cosecant function
Examples
Example 2.7.1
Evaluate ddt cosπ t.
Example 2.7.2
Evaluate ⅆⅆu cosusin2 u2.
Example 2.7.3
Evaluate ⅆⅆx sinxtanx2.
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