Math Problem Statement

Find the derivative of the trigonometric function y=-csc(x)-cos(x)

Solution

We are tasked with finding the derivative of the function:

y=csc(x)cos(x)y = -\csc(x) - \cos(x)

Step-by-Step Derivation:

  1. Derivative of csc(x)-\csc(x): The derivative of csc(x)\csc(x) is csc(x)cot(x)-\csc(x) \cot(x). So, the derivative of csc(x)-\csc(x) is: ddx[csc(x)]=csc(x)cot(x)\frac{d}{dx}[-\csc(x)] = \csc(x) \cot(x)

  2. Derivative of cos(x)-\cos(x): The derivative of cos(x)\cos(x) is sin(x)-\sin(x). So, the derivative of cos(x)-\cos(x) is: ddx[cos(x)]=sin(x)\frac{d}{dx}[-\cos(x)] = \sin(x)

Final Result:

The derivative of the function is:

dydx=csc(x)cot(x)+sin(x)\frac{dy}{dx} = \csc(x) \cot(x) + \sin(x)

Would you like more details on any part of this process?

Here are 5 related questions to deepen your understanding:

  1. What is the derivative of sec(x)\sec(x)?
  2. How is the derivative of tan(x)\tan(x) derived?
  3. Can you explain how to differentiate products of trigonometric functions?
  4. What are the applications of differentiating trigonometric functions?
  5. How can you integrate csc(x)\csc(x) and cot(x)\cot(x)?

Tip: Always remember that derivatives of trigonometric functions often involve combinations of other trigonometric functions like sin(x)\sin(x), cos(x)\cos(x), tan(x)\tan(x), etc., so it's essential to memorize these fundamental rules.

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Math Problem Analysis

Mathematical Concepts

Calculus
Trigonometric Functions
Derivatives

Formulas

Derivative of csc(x): -csc(x)cot(x)
Derivative of cos(x): -sin(x)

Theorems

Derivative rules for trigonometric functions

Suitable Grade Level

Grades 11-12 or Early College