(Q14.) So, you think you can take the derivative?

TL;DR

The second derivative of ln(secant x + tangent x) is secant x times tangent x.

Transcript

okay for number 14 uh we are once again going to find the second derivative of y is equal to ln of secant x plus tangent x okay so for the first derivative ln of something is just 1 over whatever we have right here which is secant x plus tangent x nothing change you just bring the inside down here however as usual you just multiply by the derivativ... Read More

Key Insights

• ☺️ The first derivative of ln(secant x + tangent x) is secant x.
• 🫡 To find the second derivative, differentiate the first derivative with respect to x.
• ☺️ The derivative of secant x is secant x times tangent x.
• 😑 By factoring out the common term secant x, the expression can be simplified.
• 🪈 The order of addition in the denominator does not affect the solution.
• ☺️ The second derivative of ln(secant x + tangent x) simplifies to secant x times tangent x.
• 📏 Understanding the chain rule and the derivatives of trigonometric functions is crucial for solving the problem.

Questions & Answers

Q: What is the first derivative of ln(secant x + tangent x)?

The first derivative is secant x. Explanation: The derivative of ln(secant x + tangent x) is equal to 1 divided by (secant x + tangent x), multiplied by the derivative of (secant x + tangent x), which simplifies to secant x.

Q: How do you find the second derivative of ln(secant x + tangent x)?

To find the second derivative, differentiate the first derivative with respect to x. Explanation: The derivative of secant x is secant x times tangent x. Therefore, the second derivative of ln(secant x + tangent x) is secant x times tangent x.

Q: Can the expression ln(secant x + tangent x) be simplified?

Yes, the expression can be simplified by factoring out the common term secant x from the numerator. Explanation: By factoring out secant x, the expression becomes secant x times (tangent x + secant x) divided by (secant x + tangent x). The numerator simplifies to secant x times tangent x plus secant squared x. However, in the denominator, the order of addition doesn't matter, so tangent x + secant x is the same as secant x + tangent x. These two terms cancel out, leaving the simplified expression as secant x.

Q: What is the final answer for the second derivative of ln(secant x + tangent x)?

The answer is secant x times tangent x. Explanation: After simplifying the expression and differentiating the first derivative of ln(secant x + tangent x), we get secant x times tangent x as the second derivative.

Summary & Key Takeaways

• The first derivative of ln(secant x + tangent x) is secant x.

• To find the second derivative, you need to differentiate the first derivative with respect to x.

• The derivative of secant x is secant x times tangent x.