(Q14.) So, you think you can take the derivative? | Summary and Q&A

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April 13, 2014
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blackpenredpen
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(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.

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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.

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