(Q14.) So, you think you can take the derivative?  Summary and Q&A
TL;DR
The second derivative of ln(secant x + tangent x) is secant x times tangent x.
Key Insights
 βΊοΈ The first derivative of ln(secant x + tangent x) is secant x.
 π«‘ To find the second derivative, differentiate the first derivative with respect to x.
 βΊοΈ The derivative of secant x is secant x times tangent x.
 π By factoring out the common term secant x, the expression can be simplified.
 πͺ The order of addition in the denominator does not affect the solution.
 βΊοΈ The second derivative of ln(secant x + tangent x) simplifies to secant x times tangent x.
 π Understanding the chain rule and the derivatives of trigonometric functions is crucial for solving the problem.
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
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.