derivatives of tan(x) and cot(x), quotient rule, cofunction identities | Summary and Q&A

TL;DR

The derivative of tangent X is secant squared X, while the derivative of cotangent X is negative cosecant squared X.

Key Insights

• ❎ The derivative of tangent X is secant squared X, while the derivative of cotangent X is -cosecant squared X.
• 👨‍💼 Both derivatives can be found using the quotient rule, substituting tangent X with sine X over cosine X and cotangent X with cosine X over sine X.
• 📏 Another method for finding the derivative of cotangent X is by using the derivative of tangent X and applying the chain rule.
• ❎ The derivatives of tangent and cotangent functions are always negative.
• ❓ The Pythagorean identity, sin^2 X + cos^2 X = 1, is useful in simplifying the derived formulas for the derivatives.
• 😑 The derivative of tangent X can also be expressed as secant squared X.
• ☺️ The derivative of cotangent X can be written as -cosecant squared X.

Transcript

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Questions & Answers

Q: How can I find the derivative of tangent X?

To find the derivative of tangent X, you can use the quotient rule. Replace tangent X with sine X over cosine X. Apply the quotient rule, which gives us secant squared X.

Q: What is the derivative of cotangent X?

The derivative of cotangent X can be found using the quotient rule. Replace cotangent X with cosine X over sine X. Applying the quotient rule, we get -cosecant squared X.

Q: Can I find the derivative of tangent X using a different method?

Yes, you can also find the derivative of tangent X by rewriting it as cotangent of the complementary angle (PI/2 - X). Differentiating cotangent X using the chain rule gives us -secant squared (PI/2 - X), which simplifies to secant squared X.

Q: Are the derivatives of tangent and cotangent functions always negative?

Yes, both tangent and cotangent functions have negative derivatives. This can be observed in the derived formulas for their derivatives, where secant squared X and -cosecant squared X are negative.

Summary & Key Takeaways

• The derivative of tangent X can be found using the quotient rule, which results in secant squared X.

• The derivative of cotangent X can also be found using the quotient rule, which results in negative cosecant squared X.

• Another approach to finding the derivative of cotangent X is by using the derivative of tangent X and applying the chain rule.