Worked example: Derivative of ∜(x³+4x²+7) using the chain rule

let's see if we can take the derivative with respect to X of the fourth root of x to the third power plus four x squared plus seven and at first you might say alright how do I take the derivative of fourth root of something it looks like I have a composite function I'm taking the fourth root of another expression here and you'd be right and if you're dealing with a composite function the chain rule should be front of mine but first let's just make this fourth root a little bit more tractable for us and just realize that this fourth root is really nothing but a fractional exponent so this is the same thing as the derivative with respect to X of X to the third plus four x squared plus seven to the one fourth power to the one fourth power now how do we take the derivative of this well we can view this as I said a few seconds ago we can view this as a composite function what do we do first with our X well we do all of this business and we could call this U of X and then whatever we get for U of X we raise that to the fourth power so the way that we would take the derivative we would take the derivative of this you could view it as the outer function with respect to U of X and then multiply that times the derivative of U with respect to X so let's let's do that so what this is going to be this is going to be equal to so we're going to take our outside function which I'm highlighting in green now so where I take something to the 1/4 I'm gonna take the derivative of that with respect to the inside with respect to U of X well I'm just going to use the power rule here I'm just going to bring that 1/4 out front so it's going to be 1/4 times whatever I'm taking the derivative with respect to to the 1/4 minus 1 power look all I did is use the power rule here I didn't have an X here now I'm taking the derivative with respect to U of X with respect with respect to this polynomial expression here so I could just throw the U of X in here if I like actually let me just do that so this is going to be X to the 3rd plus 4 x squared plus 7 and then I want to multiply that and this is the chain rule I took the derivative of the outside with respect to the inside and I'm going to multiply that times the derivative of the inside so what's the derivative of U of X u prime of X let's see we just going to use the power rule a bunch of times it's going to be 3x squared plus 2 times 4 is 8 X to the 2 minus 1 is just 1 power first power so that's just I can just write that as 8x and then the derivative with respect to X of 7 well derivative with respect to X of a constant it's going to be 0 so that's u prime of X so then I'm just going to multiply by u prime of X which is 3x squared plus 3x squared plus 8x and so well I can clean this up a little bit so this would be equal to this would be equal to actually let me just rewrite that exponent there so this 1/4 minus 1 I can rewrite it 1/4 minus 1 is negative three-fourths negative three-fourths negative 3/4 power and you could manipulate this in different ways if you like but the key is to just recognize that this is an application of the chain rule derivative of the outside with well actually the first thing to realize is the fourth root is the same thing as taking something to the 1/4 power basic exponent property and then realize that ok I have a composite function here so I can take the derivative of the outside with respect to the inside that's what we did here times the derivative of the inside with respect to X and so if someone were to tell you if someone were to say all right f of X f of X is equal to the fourth root of x to the third plus 4 x squared plus 7 and then they said well what is f prime of I don't know negative 3 well you would evaluate this at negative 3 so this let me just do that so it's 1/4 times C you have negative 27 I hope this works out reasonably well plus 36 plus 36 plus 7 to the negative three four it's what is this result to this is going to be equal to this is this right over here is sixteen right negative twenty seven plus seven is negative 20 plus thirty-six so this is sixteen I think this is going to work out nicely and then times 3 times negative so three times nine which is 27 minus 24 so this is going to be right over here that is going to be three now what is 16 to the negative three-fourths slimmies is 1/4 times so 16 to the 1/4 is 2 and then you raise that to the let me actually I don't want to skip steps here but this is at this point we are dealing with algebra or maybe even pre-algebra so this is going to be x times 16 to the 1/4 and then we're going to raise that to the negative 3 times that 3 out front so we could put that 3 there 16 to the 1/4 is 2 2 to the 3rd is 8 so 2 to the negative third power is 1/8 so that is 1/8 so we have 3/4 times 1/8 which is equal to 3 over 32 3/32 so that would be the slope of the tangent line of the graph y is equal to f of X when X is equal to negative 3