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

Previous problemNext problem

2:07 minutes

Problem 3.5.23

Textbook Question

Find the derivative of the following functions.
y = sin x + cos x

Verified step by step guidance

Identify the function for which you need to find the derivative: $ y = \sin x + \cos x $.

Recall the basic derivatives of trigonometric functions: $ \frac{d}{dx}(\sin x) = \cos x $ and $ \frac{d}{dx}(\cos x) = -\sin x $.

Apply the sum rule for derivatives, which states that the derivative of a sum of functions is the sum of their derivatives.

Differentiate each term separately: $ \frac{d}{dx}(\sin x) = \cos x $ and $ \frac{d}{dx}(\cos x) = -\sin x $.

Combine the results to find the derivative of the entire function: $ \frac{dy}{dx} = \cos x - \sin x $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that represents the slope of the tangent line to the curve of the function at any given point. Derivatives are used to find rates of change and can be computed using various rules and techniques.

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. In calculus, these functions have specific derivatives: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Understanding these derivatives is essential for differentiating functions involving trigonometric expressions.

Sum Rule of Derivatives

The sum rule of derivatives states that the derivative of a sum of functions is equal to the sum of their derivatives. This means that if you have a function that is the sum of two or more functions, you can differentiate each function separately and then add the results together. This rule simplifies the process of finding derivatives for functions like y = sin x + cos x.