Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

Derivatives of Trig Functions

Calculus

3. Techniques of Differentiation

Derivatives of Trig Functions

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1:40 minutes

Problem 3.5.42

Textbook Question

23–51. Calculating derivatives Find the derivative of the following functions.
y = tan x + cot x

Verified step by step guidance

Step 1: Identify the functions involved in the expression. The given function is \( y = \tan x + \cot x \). Here, \( \tan x \) and \( \cot x \) are trigonometric functions.

Step 2: Recall the derivatives of the basic trigonometric functions. The derivative of \( \tan x \) is \( \sec^2 x \), and the derivative of \( \cot x \) is \( -\csc^2 x \).

Step 3: Apply the sum rule for derivatives. The sum rule states that the derivative of a sum of functions is the sum of their derivatives. Therefore, \( \frac{d}{dx}(\tan x + \cot x) = \frac{d}{dx}(\tan x) + \frac{d}{dx}(\cot x) \).

Step 4: Substitute the derivatives of the individual functions into the expression. This gives \( \sec^2 x - \csc^2 x \).

Step 5: Simplify the expression if possible. In this case, the expression \( \sec^2 x - \csc^2 x \) is already in its simplest form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate at which a function changes at any given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, are functions of an angle that relate the angles of a triangle to the lengths of its sides. In calculus, understanding the derivatives of these functions is crucial, as they often appear in various mathematical models and applications.

Derivative Rules

Derivative rules, such as the sum rule, product rule, and quotient rule, provide systematic methods for finding the derivatives of complex functions. The sum rule states that the derivative of a sum of functions is the sum of their derivatives, which is particularly useful when differentiating functions like y = tan x + cot x.

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Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

Find the velocity of the oscillator, v(t) =y′(t).

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An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

At what times and positions is the velocity zero?

Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

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23–51. Calculating derivatives Find the derivative of the following functions.y = tan x + cot x

Key Concepts

Derivatives

Trigonometric Functions

Derivative Rules

Watch next

23–51. Calculating derivatives Find the derivative of the following functions.
y = tan x + cot x