Find the derivative of Y with respect to X for Y equal hyperbolic sine inverse of three square roots of 15X. So the first thing we need to know is we need to figure out what the derivative of just hyperbolic sine is. If we don't recall, start by letting F of X equal hyperbolic sine so that the inverse is hyperbolic sine inverse of X. Then use our official definition of A inverse derivative, which says the derivative of the inverse equals 1 divided by the derivative of the original function. We know that the derivative of hyperbolic sine is hyperbolic cosine sticking in the inverse function, which is the hyperbolic sine inverse of X. Now we have a relationship for our hyperbolics that says hyperbolic cosine squared X minus hyperbolic sine squared X equal 1. So if we solve for hyperbolic cosine X, we can see that that's equaling square root 1 plus hyperbolic sine squared X. So taking out the hyperbolic cosine, we're going to put in one divided by the square root 1 plus hyperbolic sine squared of hyperbolic sine inverse of X, and then the hyperbolic sine squared and the hyperbolic sine inverse is going to really cancel, leaving us just X ^2. So now this is the formula we're going to use, but instead of X, we're going to have it be 3 square roots of 15X. So we get 1 divided by the square root 1 + 3 square roots of 15 X squared. Then we have to multiply that by the derivative of three square roots of 15X, so the derivative of three square roots of 15X. Remember, the derivative of a square root is just 1 / 2 of whatever is on the inside of the square root. So that three is a constant, it stays on top. But now we need to take the derivative of the inside. So we're going to multiply by the derivative of the 15X derivative of the 15X is 15. So now when we simplify this all up, three square roots of 15 X squared is 135 X 3 * 15 is 45. So we get 45 / 2 square roots, fifteen X + 2025 X squared.