2.9 Calculus of Hyperbolic Functions

[STARTS AT 2:14]

INSTRUCTOR: All right, so let's find a derivative of hyperbolic sine of x squared. OK? Now, we have to keep in mind, we have to worry about the u substitution or our chain rule here, OK? So the outside function is the sine, hyperbolic sine. So the derivative of that is the hyperbolic cosine of x squared times the derivative over inside function, which is going to be 2x, OK? So this is 2x hyperbolic cosine of x squared. OK? 

Same idea with this one, except now our inside function is the hyperbolic cosine. All right, so our chain rule would say this is 2 times the hyperbolic cosine x. I guess I don't really need to write that there, but there's parentheses. Hyperbolic cosine of x to the first power times the derivative of hyperbolic cosine, which is hyperbolic sine x. 

Next, derivative of tangent of x squared plus 3x. OK, well, the outside function, the derivative of that is hyperbolic secant squared. We'll keep the inside function times the derivative of the inside function, which in this case is 2x plus 3. 

Next, we have the derivative of 1 over hyperbolic sine squared. OK, now, I wrote it this way with the parentheses squared rather than with the 2 here just to emphasize the fact that that is an inside function, OK? We have a u substitution. So let's write this as the derivative of sine hx to the negative 2. Let me add parentheses there. 

In that case, the derivative will be cosine, hyperbolic cosine. We'll bring a negative 2 down. This is to the-- actually, we don't need that. I got a little ahead of myself. Hold on. We need to keep our inside functions. So negative 2 hyperbolic sine to the negative 3 times the derivative hyperbolic cosine. So this will end up being negative 2 hyperbolic cosine over hyperbolic sine cubed.

[ENDS AT 5:00]