Differential Calculus - Chain Rule for Trigonometric Functions

Summary

This video covers the chain rule in differential calculus, specifically focusing on trigonometric functions. It explains how to differentiate composite trigonometric functions like sine, cosine, tangent, secant, cotangent, and cosecant using the chain rule. The instructor walks through several examples, highlighting common mistakes and emphasizing the importance of parentheses in simplifying the results. By the end, viewers learn how to differentiate functions within functions, apply the chain rule, and simplify expressions accurately. The video provides clear explanations for students to better understand and apply these concepts.

Takeaways

• 📘 The video covers the chain rule for trigonometric functions in differential calculus.
• 📌 The derivative of sin(U) is equivalent to cos(U) multiplied by U'.
• 🔍 The derivative of tan(U) is sec^2(U) multiplied by U'.
• ✏️ The derivative of cos(U) is -sin(U) multiplied by U'.
• 📐 For cot(U), the derivative is -csc^2(U) multiplied by U'.
• 📈 The chain rule is applied by differentiating the outer function, then multiplying by the derivative of the inner function (U').
• 📝 Example 1: For y = sin(2x), the derivative is 2 cos(2x).
• 🔑 Example 2: For y = cos(x-1), the derivative is -sin(x-1).
• 📊 Example 3: For y = tan(3x), the derivative is 3 sec^2(3x).
• ✅ The key takeaway is understanding how to apply the chain rule effectively for differentiating composite trigonometric functions.

Q & A

What is the chain rule in the context of trigonometric functions?

The chain rule for trigonometric functions involves differentiating the outer function normally and then multiplying by the derivative of the inner function (U), which is referred to as U prime (U').

What is the derivative of sin(U) using the chain rule?

The derivative of sin(U) using the chain rule is cos(U) multiplied by U'.

How do you apply the chain rule to differentiate tan(U)?

To differentiate tan(U) using the chain rule, the result is sec²(U) multiplied by U'.

What is the formula for differentiating sec(U) using the chain rule?

The derivative of sec(U) using the chain rule is sec(U)tan(U) multiplied by U'.

What happens when you differentiate cos(U) using the chain rule?

When differentiating cos(U) using the chain rule, the result is -sin(U) multiplied by U'.

How do you find the derivative of cot(U) using the chain rule?

The derivative of cot(U) using the chain rule is -csc²(U) multiplied by U'.

What steps should be followed when applying the chain rule to differentiate y = sin(2x)?

First, identify the inner function U = 2x. Then, differentiate sin(U) to get cos(2x) and multiply it by the derivative of U, which is 2. The final answer is 2cos(2x).

How do you apply the chain rule to y = cos(x - 1)?

First, identify U = x - 1. Then, differentiate cos(U) to get -sin(x - 1) and multiply it by the derivative of U, which is 1. The final answer is -sin(x - 1).

What is the result of differentiating y = tan(3x) using the chain rule?

First, identify U = 3x. Differentiate tan(U) to get sec²(3x) and multiply it by the derivative of U, which is 3. The final answer is 3sec²(3x).

How do you differentiate y = sec(5x) using the chain rule?

First, identify U = 5x. Differentiate sec(U) to get sec(5x)tan(5x) and multiply by the derivative of U, which is 5. The final answer is 5sec(5x)tan(5x).