Video Transcript
Find the derivative of the function 𝐹 of 𝑡 equals the natural log of the hyperbolic
sin of 𝑡, or sinh of 𝑡.
Recall that if we have the natural log of another function 𝑓 of 𝑡, then the
derivative is just the derivative of the function 𝑓 prime of 𝑡 over the original
function 𝑓 of 𝑡. So in this case, d by d𝑡 of the natural log of sinh 𝑡 is just equal to d by d𝑡 of
sinh 𝑡 over sinh 𝑡. d by d𝑡 of sinh 𝑡 is just equal to cosh 𝑡. So this gives us cosh 𝑡 over sinh 𝑡. We have the hyperbolic identity tanh 𝑡 is identically equal to sinh 𝑡 over cosh
𝑡. Our result here is just the reciprocal of tanh 𝑡. So we have one over tanh 𝑡.
Now also recall that the hyperbolic cotan or coth of 𝑡 is identically equal to one
over the tanh of 𝑡. So this gives us our final answer. 𝐹 prime of 𝑡 is equal to coth of 𝑡.
