Introduction to Trigonometric Derivatives

Trigonometric functions are fundamental tools that keep showing up throughout mathematics, especially in a course like Calculus 4B. Because they recur “again and again,” a solid grasp of their basic derivatives is a prerequisite for success in later classes. The instructor, Mr. Leonard, begins by promising a proof of the derivative of sin x using the limit definition of a derivative.

Proof of the Derivative of sin x

Recall the definition of a derivative as a limit:

[ \frac{d}{dx}f(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}. ]

Applying this to f(x)=sin x requires the identity

[ \sin(x+h)=\sin x\cos h+\cos x\sin h. ]

Substituting and separating the limits gives

[ \frac{d}{dx}\sin x = \sin x\;\lim_{h\to0}\frac{\cos h-1}{h} + \cos x\;\lim_{h\to0}\frac{\sin h}{h}. ]

The two limits are well‑known:

[ \lim_{h\to0}\frac{\cos h-1}{h}=0,\qquad \lim_{h\to0}\frac{\sin h}{h}=1. ]

Therefore

[ \frac{d}{dx}\sin x = \cos x. ]

The instructor emphasizes the “weird” fact that plugging a number into cos x gives the slope of sin x at that point.

Derivatives of the Other Trigonometric Functions

Using similar limit arguments or the quotient rule, the remaining basic derivatives follow a tidy pattern:

These results can be memorized in a table; the patterns between the “co‑functions” (secant/cosecant and tangent/cotangent) often involve a negative sign.

Applications of Derivatives

A derivative tells us the instantaneous slope of a curve, which in turn lets us write the equation of a tangent line. For example, consider

[ y = x\sin x. ]

Using the product rule,

[ y' = \sin x + x\cos x. ]

To find the tangent line at (x = \pi/2):

• Slope (m = \sin(\pi/2) + (\pi/2)\cos(\pi/2) = 1 + 0 = 1).
• Point ((\pi/2,\; y(\pi/2)) = (\pi/2,\; (\pi/2)\sin(\pi/2)) = (\pi/2,\; \pi/2)).

The tangent line equation is

[ y - \frac{\pi}{2} = 1\bigl(x - \frac{\pi}{2}\bigr) \quad\Longrightarrow\quad y = x. ]

Thus the line (y = x) touches the curve exactly at that point.

Practice Problems

1. Derivative of (\displaystyle \frac{\sin x}{1+\cos x})
   Apply the quotient rule:

[ \frac{d}{dx}\Bigl(\frac{\sin x}{1+\cos x}\Bigr) = \frac{(1+\cos x)\cos x - \sin x(-\sin x)}{(1+\cos x)^2} = \frac{1}{1+\cos x}, ]

after simplifying with (\sin^2 x + \cos^2 x = 1).

1. Repeated derivatives of sin x
   The sequence cycles every four steps:

[ \sin x,\; \cos x,\; -\sin x,\; -\cos x,\; \sin x,\dots ]

Consequently, the fourth derivative of sin x is again sin x.

Real‑World Application: Oscillating Spring‑Mass System

A simple harmonic oscillator can be modeled with a cosine function. If a spring is pulled 3 cm beyond its equilibrium and released at (t=0), the position can be written as

[ P(t) = -3\cos t. ]

The velocity is the first derivative:

[ v(t) = \frac{d}{dt}[-3\cos t] = 3\sin t. ]

Thus the velocity at any instant is directly given by a sine function scaled by the amplitude.

Introduction to the Chain Rule

When functions become compositions—such as a large exponent applied to a trig function—the basic product and quotient rules become cumbersome. The chain rule treats the outer and inner functions separately, allowing us to differentiate efficiently. It will be the final differentiation rule taught, tying together all the earlier techniques and preparing students for more advanced calculus topics.