One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.
We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function f ( x ) , f ( x ) ,
f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h . f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h .Consequently, for values of h h very close to 0, f ′ ( x ) ≈ f ( x + h ) − f ( x ) h . f ′ ( x ) ≈ f ( x + h ) − f ( x ) h . We see that by using h = 0.01 , h = 0.01 ,
d d x ( sin x ) ≈ sin ( x + 0.01 ) − sin x 0.01 d d x ( sin x ) ≈ sin ( x + 0.01 ) − sin x 0.01By setting D ( x ) = sin ( x + 0.01 ) − sin x 0.01 D ( x ) = sin ( x + 0.01 ) − sin x 0.01 and using a graphing utility, we can get a graph of an approximation to the derivative of sin x sin x (Figure 3.25).
Upon inspection, the graph of D ( x ) D ( x ) appears to be very close to the graph of the cosine function. Indeed, we will show that
d d x ( sin x ) = cos x . d d x ( sin x ) = cos x .If we were to follow the same steps to approximate the derivative of the cosine function, we would find that
d d x ( cos x ) = − sin x. d d x ( cos x ) = − sin x.The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.
d d x ( sin x ) = cos x d d x ( sin x ) = cos x d d x ( cos x ) = − sin x d d x ( cos x ) = − sin xBecause the proofs for d d x ( sin x ) = cos x d d x ( sin x ) = cos x and d d x ( cos x ) = − sin x d d x ( cos x ) = − sin x use similar techniques, we provide only the proof for d d x ( sin x ) = cos x . d d x ( sin x ) = cos x . Before beginning, recall two important trigonometric limits we learned in Introduction to Limits:
lim h → 0 sin h h = 1 and lim h → 0 cos h − 1 h = 0 . lim h → 0 sin h h = 1 and lim h → 0 cos h − 1 h = 0 .
The graphs of y = ( sin h ) h y = ( sin h ) h and y = ( cos h − 1 ) h y = ( cos h − 1 ) h are shown in Figure 3.26.
Figure 3.26 These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.
We also recall the following trigonometric identity for the sine of the sum of two angles:
sin ( x + h ) = sin x cos h + cos x sin h . sin ( x + h ) = sin x cos h + cos x sin h .Now that we have gathered all the necessary equations and identities, we proceed with the proof.
d d x sin x = lim h → 0 sin ( x + h ) − sin x h Apply the definition of the derivative. = lim h → 0 sin x cos h + cos x sin h − sin x h Use trig identity for the sine of the sum of two angles. = lim h → 0 ( sin x cos h − sin x h + cos x sin h h ) Regroup. = lim h → 0 ( sin x ( cos h − 1 h ) + cos x ( sin h h ) ) Factor out sin x and cos x . = sin x · 0 + cos x · 1 Apply trig limit formulas. = cos x Simplify. d d x sin x = lim h → 0 sin ( x + h ) − sin x h Apply the definition of the derivative. = lim h → 0 sin x cos h + cos x sin h − sin x h Use trig identity for the sine of the sum of two angles. = lim h → 0 ( sin x cos h − sin x h + cos x sin h h ) Regroup. = lim h → 0 ( sin x ( cos h − 1 h ) + cos x ( sin h h ) ) Factor out sin x and cos x . = sin x · 0 + cos x · 1 Apply trig limit formulas. = cos x Simplify.
Figure 3.27 shows the relationship between the graph of f ( x ) = sin x f ( x ) = sin x and its derivative f ′ ( x ) = cos x . f ′ ( x ) = cos x . Notice that at the points where f ( x ) = sin x f ( x ) = sin x has a horizontal tangent, its derivative f ′ ( x ) = cos x f ′ ( x ) = cos x takes on the value zero. We also see that where f ( x ) = sin x f ( x ) = sin x is increasing, f ′ ( x ) = cos x > 0 f ′ ( x ) = cos x > 0 and where f ( x ) = sin x f ( x ) = sin x is decreasing, f ′ ( x ) = cos x < 0 . f ′ ( x ) = cos x < 0 .
Figure 3.27 Where f ( x ) f ( x ) has a maximum or a minimum, f ′ ( x ) = 0 f ′ ( x ) = 0 that is, f ′ ( x ) = 0 f ′ ( x ) = 0 where f ( x ) f ( x ) has a horizontal tangent. These points are noted with dots on the graphs.
