Math Problem Statement
Show that the derivative of cot(x) is -csc^2(x).
Solution
To show that:
Step-by-step differentiation:
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Express : Now we need to differentiate this using the quotient rule.
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Quotient Rule: If , then: Here, and .
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Differentiate and :
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Apply the quotient rule: Simplifying the numerator:
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Use the Pythagorean identity: Therefore, the expression becomes:
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Simplify:
Thus, we have shown that:
Would you like more details on any part of this solution?
Related questions:
- How do you differentiate other trigonometric functions like or ?
- What is the derivative of , and how does it compare to ?
- Can you apply the quotient rule to non-trigonometric functions?
- How does the chain rule apply to trigonometric functions?
- What are the integration counterparts of these trigonometric derivatives?
Tip:
Memorizing the derivatives of basic trigonometric functions will save you a lot of time in calculus problems.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Calculus
Derivatives
Formulas
Quotient Rule
Derivative of sin(x)
Derivative of cos(x)
Pythagorean Identity
Theorems
Quotient Rule
Pythagorean Identity: sin^2(x) + cos^2(x) = 1
Suitable Grade Level
Grades 11-12 (A-level Calculus)