Derivatives of the Trigonometric Functions
Derivative of f(x) = sin(x) We begin by exploring an important limit. We can graph the function
sin x The graph is given below
The graph clearly indicates that as x approaches 0, y approaches 1. We will use this fact in to find the derivative of f(x) = sin x. The fact can be proven using triangles, but that should be done in a more advanced course. Another important limit that help in the proof is
Once again, we will not prove this fact. It can be found in more advanced textbooks.
Find the derivative of f(x) = x^{2} sin(3x)
Solution We use the product and the chain rules. f '(x) = (2x)(sin(3x)) + (x^{2})(3cos(3x)) = 2x sin(3x) + 3x^{2} cos(3x)
The Other Trigonometric Functions We can find the derivatives of the other five trigonometric functions by using trig identities and rules of differentiation. Below is a list of the six trig functions and their derivatives.
We will prove two of these.
The proof to the derivative of cos x We use the identity cos x = sin(p/2  x) Take the derivative of both sides using the chain rule for the derivative of the right hand side. (cos x)' =  cos(p/2  x) Now use the identity sin x = cos(p/2  x) to arrive at the result (cos x)' =  sin x The proof to the derivative of tan x We use the quotient rule on
sin x
(cos x)(cos x)  (sin x)(sin x)
cos^{2} x + sin^{2}
x
1
Application The amount A of food available at day t of the year for a North American hare can be modeled by the equation
2p(t  120) On what day does the hare have the most food available?
Solution We are asked to find a maximum, which involves taking the first derivative and setting it equal to zero. We have
320(2p)
2p(t  120) Setting it equal to zero and solving gives us
2p(t  120) When k = 0, we get t = 120 and when k = 1 we get t = 302.5. The second derivative test will tell us that t = 120 gives us a maximum and t = 320 gives us a minimum. Other values of k will give us the same day of the year. We can conclude that the hare has the greatest food supply on April 30.
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