in this lesson we're going to focus on differentiating inverse trigonometric functions so let's get right into it what is the derivative of arc sine of x cubed go ahead and try this problem now you need to know the formula if you know the formula life is easy the derivative of some generic arc sine function let's say arc sine u is equal to u prime divided by the square root of 1 minus u squared so all you need to do is identify u and u prime well the first part is straightforward because u is the stuff inside of the arc sine function so we can see that u is equal to x cubed now u prime the derivative of u that's going to be the derivative of x cubed which is 3x squared so now we can get the answer so the derivative of arc sine x cubed is going to be u prime which is 3x squared divided by the square root of 1 minus u squared or 1 minus x cubed squared so the final answer is 3x squared divided by the square root of 1 minus x to the sixth power and so this is the answer now let's try another example what is the derivative of arc cosine 5x minus 9. go ahead and try that so here's the formula that you need the derivative of arc cosine of u is very similar to the arc sine of u formula it's going to be negative u prime instead of positive u prime divided by the square root of 1 minus u squared so in this example u is five x minus nine that's the stuff inside of the r cosine function and the derivative of five x minus nine is simply five so the answer is gonna be negative five divided by the square root of 1 minus 5x minus 9 squared and that's the solution for this problem here's another problem that you could work with what is the derivative of arc tangent square root x so if you don't have the formula check out your textbook it should be in it but here it is for arc tangent the derivative of arc tan or inverse tan of u is going to be u prime divided by one plus u squared so we can see that u is the square root of x which if we rewrite it that's x to the one half now u prime that's i brought you to the half for some reason u prime is the derivative of x to the one half so using the power rule it's going to be one half x to the minus one half and we can rewrite that as one over two x to the positive one half and then converting it back to radical form it's one over two square root x so now let's use the formula it's u prime which is square root x divided by 1 plus well actually i take that back u prime is this thing right here so that's 1 over 2 square root x and then divided by 1 plus u squared so u is the square root of x and once we square it it simply becomes x so this is 1 over 2 square root x divided by 1 plus x now we could simplify the expression now let's multiply the top and the bottom by 2 square root x so we can get rid of the fraction in the numerator so this is going to be one over one plus x times two square root x and if you need to rationalize the expression you can multiply the top and bottom by the square root of x so on the denominator the square root of x times the square root of x simply is x so you can leave your final answer like that so it's the square root of x divided by two x times one plus x and so that's the inverse tangent or the derivative of arctangent of square root x let's consider one more example find the derivative of arc secant x to the fourth power so let's begin with the formula so the derivative of arc secant is going to be u prime divided by the absolute value of u times the square root of u squared minus one so we can see that u is x to the fourth and u prime is four x cubed so this is going to be u prime which is 4 x cubed divided by the absolute value of u which is x to the fourth now because this term will always be positive we really don't need the absolute value around it and then it's going to be the square root of u squared so x to the 4 squared is really x to the 8 and then minus 1. now we could simplify this expression we could cancel three of the four x variables leaving one in a denominator so this is going to be four and this time i'm going to put back the absolute value symbol so four over x times the square root of x to the eight minus one and so that's the derivative of arc secant x to the fourth power you