we want to determine the derivative of each of the given functions notice how they're all exponential functions with base e and the function f ofx equal e to the x is special for many reasons but one of them is that the function is its own derivative the derivative of e to the X with respect to X is just e to the X however we do have to be a little bit careful here if the exponent here is a function of X other than just X we do have to apply the chain rule the dtive of e to theu U with respects to X is e to U * U Prime so this formula here has the chain rule built in so looking at these three functions again notice how we will have to apply the chain rule to determine each derivative so for this first example f ofx = e ^ of 3x the exponent or U would be equal to 3x therefore U Prime would be equal to 3 so now we can rewrite this function in terms of of U as e to the U so the derivative would just be e to the U * U prime or E to the^ 3x * 3 we can rewrite this as 3 e 3x in the next example we have F ofx = 16 e^ of x^2 so in this case we would have U = to x^2 so U Prime would be equal to 2x so now we can think of this as 1 16 e to the U so our derivative would be equal to6 e to the U * U prime or e to the x^2 power * 2x now if we simplify this notice we'd have 16 * * 2 that would be 26 or 1/3 x e^ of x^2 and then for our last example we have F ofx = 4 * e ^ of < TK X so here U would be equal to theun of X but in order to determine U Prime we'd write this as X the^ of 12 remember we have an index of two and an exponent of one so U Prime would be 12 * x ^ of - 12 and then simplifying this we'd have 1 all over 2 x the2 in the denominator which we can write as the square of X so now we can think of this as 4 * e to the U so frime of x would be equal to 4 * the D of e to the U which is just e to the U * U prime or e to the < TK X power times 1 all over 2 < TK X so now we'll go ahead and simplify this well 4 * 1/2 would be 2 so we'd have 2 e^ of < TK x / X so the main thing to remember here is if the exponent is a function of X other than just X we do have to apply the chain rule given by the second formula we'll take a look at several more examples in the next video