Derivative Of Trigonometric Functions Worksheet

Mastering Trigonometric Derivatives: A Comprehensive Worksheet and Guide

This worksheet and accompanying guide will help you master the derivatives of trigonometric functions. Understanding these derivatives is crucial for success in calculus and beyond, forming the foundation for many advanced applications in physics, engineering, and other fields. We'll cover the fundamental derivatives, explore their proofs (where appropriate), and provide numerous practice problems to solidify your understanding. This guide assumes a basic understanding of differentiation and trigonometric identities. Let's begin!

I. Introduction to Trigonometric Derivatives

Trigonometric functions—sine (sin x), cosine (cos x), tangent (tan x), cotangent (cot x), secant (sec x), and cosecant (csc x)—describe relationships between angles and sides in triangles. Their derivatives, however, reveal the rate of change of these functions. This is incredibly important in various applications because it allows us to analyze how angles and lengths change with respect to other variables.

The derivatives of these functions aren't intuitive; they require understanding of limits and the unit circle. However, once learned, they become straightforward tools in your calculus arsenal.

II. The Fundamental Derivatives

Memorizing these derivatives is essential. We'll follow with explanations and proofs where applicable.

• Derivative of sin x: d(sin x)/dx = cos x
• Derivative of cos x: d(cos x)/dx = -sin x
• Derivative of tan x: d(tan x)/dx = sec²x
• Derivative of cot x: d(cot x)/dx = -csc²x
• Derivative of sec x: d(sec x)/dx = sec x tan x
• Derivative of csc x: d(csc x)/dx = -csc x cot x

III. Proofs of Selected Derivatives (for deeper understanding)

Let's prove the derivatives of sin x and cos x using the limit definition of a derivative and some trigonometric identities. Understanding these proofs enhances your comprehension and allows you to derive the others using the quotient rule (for tan x, cot x, sec x, and csc x).

A. Proof of d(sin x)/dx = cos x

The limit definition of a derivative is:

f'(x) = lim (h→0) [(f(x + h) - f(x))/h]

For f(x) = sin x:

f'(x) = lim (h→0) [(sin(x + h) - sin x)/h]

Using the trigonometric identity sin(A + B) = sin A cos B + cos A sin B:

f'(x) = lim (h→0) [(sin x cos h + cos x sin h - sin x)/h]

Rearranging:

f'(x) = lim (h→0) [sin x (cos h - 1)/h + cos x (sin h)/h]

We know from limit properties:

lim (h→0) (sin h)/h = 1 lim (h→0) (cos h - 1)/h = 0

Therefore:

f'(x) = sin x * 0 + cos x * 1 = cos x

B. Proof of d(cos x)/dx = -sin x

Following a similar process:

f'(x) = lim (h→0) [(cos(x + h) - cos x)/h]

Using the trigonometric identity cos(A + B) = cos A cos B - sin A sin B:

f'(x) = lim (h→0) [(cos x cos h - sin x sin h - cos x)/h]

Rearranging:

f'(x) = lim (h→0) [cos x (cos h - 1)/h - sin x (sin h)/h]

Using the limits we established earlier:

f'(x) = cos x * 0 - sin x * 1 = -sin x

IV. Chain Rule with Trigonometric Functions

The chain rule is essential when differentiating composite functions. Remember the chain rule: d(f(g(x)))/dx = f'(g(x)) * g'(x).

Let's look at some examples:

• Find the derivative of sin(2x):

Here, f(x) = sin x and g(x) = 2x. Then f'(x) = cos x and g'(x) = 2.

Applying the chain rule: d(sin(2x))/dx = cos(2x) * 2 = 2cos(2x)

• Find the derivative of cos(x²):

Here, f(x) = cos x and g(x) = x². Then f'(x) = -sin x and g'(x) = 2x.

Applying the chain rule: d(cos(x²))/dx = -sin(x²) * 2x = -2x sin(x²)

• Find the derivative of tan(3x + 1):

Here, f(x) = tan x and g(x) = 3x + 1. Then f'(x) = sec²x and g'(x) = 3.

Applying the chain rule: d(tan(3x + 1))/dx = sec²(3x + 1) * 3 = 3sec²(3x + 1)

V. Worksheet: Practice Problems

Now, let's put your knowledge into practice. Solve the following problems, showing your work. Solutions are provided at the end.

Part 1: Basic Derivatives

1. Find the derivative of y = sin x + cos x
2. Find the derivative of y = 3tan x – 2sec x
3. Find the derivative of y = 5cot x + 4csc x
4. Find the derivative of y = 2sin x - 4cos x + 7tan x
5. Find the derivative of y = sin x * cos x (Hint: use the product rule)
6. Find the derivative of y = sin x / cos x

Part 2: Chain Rule

1. Find the derivative of y = sin(3x)
2. Find the derivative of y = cos(x³)
3. Find the derivative of y = tan(2x – 5)
4. Find the derivative of y = sec(√x)
5. Find the derivative of y = cos²(x)
6. Find the derivative of y = sin(e^x)

Part 3: More Complex Problems

1. Find the derivative of y = x²sin x
2. Find the derivative of y = (cos x) / (1 + sin x)
3. Find the derivative of y = sin(x²) cos(2x)
4. Find the second derivative of y = sin(2x)
5. Find the equation of the tangent line to y = cos x at x = π/2

VI. Solutions to Worksheet Problems

Part 1:

1. dy/dx = cos x - sin x
2. dy/dx = 3sec²x - 2sec x tan x
3. dy/dx = -5csc²x - 4csc x cot x
4. dy/dx = 2cos x + 4sin x + 7sec²x
5. dy/dx = cos²x - sin²x
6. dy/dx = sec²x

Part 2:

1. dy/dx = 3cos(3x)
2. dy/dx = -3x²sin(x³)
3. dy/dx = 2sec²(2x - 5)
4. dy/dx = (sec(√x)tan(√x))/(2√x)
5. dy/dx = -2cos x sin x = -sin(2x)
6. dy/dx = e^x cos(e^x)

Part 3:

1. dy/dx = 2xsin x + x²cos x
2. dy/dx = -1
3. dy/dx = 2x cos(x²)cos(2x) - 2sin(x²)sin(2x)
4. d²y/dx² = -4sin(2x)
5. The tangent line is y = -x + π/2

VII. Frequently Asked Questions (FAQ)

Q: Why is the derivative of cos x negative?

A: The negative sign arises from the way the cosine function decreases as the angle increases in the first quadrant of the unit circle. The derivative reflects this decreasing nature.

Q: How can I remember all these derivatives?

A: Consistent practice and using mnemonics can help. Try writing them out repeatedly, creating flashcards, or developing a personal mnemonic device to aid recall.

Q: What are some common mistakes to avoid?

A: Common mistakes include forgetting the chain rule, misapplying trigonometric identities, and incorrect use of the quotient rule. Pay close attention to detail and practice regularly to minimize these errors.

Q: Where are these derivatives used in real-world applications?

A: Trigonometric derivatives are fundamental in physics (analyzing oscillations, waves), engineering (designing curves, analyzing motion), and computer graphics (modeling curves and surfaces).

VIII. Conclusion

Mastering the derivatives of trigonometric functions is a cornerstone of calculus. Through understanding the fundamental derivatives, their proofs, and applications of the chain rule, you've significantly enhanced your calculus skills. Consistent practice, working through problems, and reviewing the concepts will solidify your understanding and prepare you for more advanced calculus topics. Remember, the key is consistent effort and a persistent curiosity about how these concepts function. Keep practicing, and you'll be well on your way to mastering calculus!