derivative of ln(sinh(x)) vs derivative of sinh(ln(x))

TL;DR

This content explains the process of differentiating hyperbolic functions and logarithms, providing step-by-step explanations and examples.

Transcript

okay we're two terrific questions on the spot so the first one see the relative of enough scene checks and for the second one yes we just switch the order we have to take the derivative of change of Ln X so as always please pause the video and do the easier one first alright this right here it's actually easier because he only takes two lines so he... Read More

Key Insights

• 😑 When differentiating "cosh" of a natural logarithm, we can either directly apply the chain rule or simplify the expression using the identity for "cosh".
• ☺️ The derivative of "cosh" of Ln x is hyperbolic cotangent x.
• 😒 For differentiating "sinh" of a natural logarithm, we can use the same methods as for "cosh", with hyperbolic cosine as the derivative.
• ☺️ The derivative of "sinh" of Ln x is (x - 1/x)/2 or (x^2 - 1)/(2x^2), depending on whether we simplify or differentiate directly.

Questions & Answers

Q: How do you differentiate the hyperbolic function "cosh" of a natural logarithm?

To differentiate "cosh" of Ln x, we apply the chain rule, which states that we need to multiply the derivative of Ln x (which is 1/x) with the derivative of "cosh" (which is hyperbolic cotangent) of x. The final result is hyperbolic cotangent x.

Q: Is there another method to differentiate "cosh" of a natural logarithm?

Yes, another method is to simplify the expression first. By using the identity for "cosh" (cosh x = (e^x + e^(-x))/2), we can substitute Ln x into the expression and simplify to get x + 1/x. Then, we differentiate this simplified expression, which gives us (x^2 + 1)/(2x^2).

Q: How do you differentiate the hyperbolic function "sinh" of a natural logarithm?

To differentiate "sinh" of Ln x, we use the chain rule. The derivative of "sinh" is hyperbolic cosine (cosh), and we multiply it with the derivative of Ln x (which is 1/x). The final result is (x - 1/x)/2.

Q: Can we use a simplified method to differentiate "sinh" of a natural logarithm?

Yes, we can simplify the expression first by using the identity for "sinh" (sinh x = (e^x - e^(-x))/2). By substituting Ln x into the expression and simplifying, we get x - 1/x. Differentiating this simplified expression gives us (x^2 - 1)/(2x^2).

Summary & Key Takeaways

• The video explains how to differentiate the hyperbolic function "cosh" of a natural logarithm.

• The video also demonstrates another method of differentiating "cosh" of a natural logarithm by simplifying the expression first.

• The content covers the differentiation of the hyperbolic function "sinh" of a natural logarithm, showing both the direct method and the simplified method.