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given capital f of X is equal to G of X to the third power where the graph of G and it's tangent line of x equals four are shown what is the value of f prime of 4 so they give us G of X right over here in blue and they also show us the slope they show us the tangent line at x equals 4 right over here so we need to figure out f prime of 4 so let's just rewrite this information they've given us we know that f of X f of X is equal to G of X to the third power so I'll write it like this G of X G of X to the third power so we want to figure out what f prime of X is when X is equal to 4 so let's just take the derivative here of both sides with respect to X so take the derivative of the left-hand side with respect to X and take the derivative of the right-hand side with respect to X so the left-hand side this is just going to be capital F prime of X capital F prime of X now on the right hand side I have a composite of G of X to the third power so first we can view this as the product of the derivative of G of X to the third power with respect to G of X so there we could literally just apply what we know about the power rule if the derivative of x to the third is 3x the derivative of X to the third with respect to X is 3x squared so the derivative of GX to the third with respect to G of X is just going to be 3 times G of X to the second power and they're going to multiply that times the derivative of G of X with respect to X so times G prime of X and this comes straight out of the chain rule derivative of this it's derivative G of X to the third with respect to G of X which is this times the derivative of G of X with respect to X which is that right over right over there so now let's just substitute we want we want to figure out what this derivative is when X is equal to 4 so we could say that F prime of 4 is equal to 3 times G of four squared times G prime of four so what is G of four what is G of four going to be well we can just look at our function right over here when x equals four our function is equal to three our function is equal to three so G of four is equal to three and what's G prime of four so when our when x equals four G prime of 4 is the slope of the tangent line and they've drawn us they've drawn the tangent line when x equals four here so what is the slope of this line so we just have to think about change in Y over change in X and I'll look at that between two integer valued coordinates so liquid looks like between these two points and when we increase X by two we decrease Y we decrease Y by four so as you remember slope is rise over run or change in Y over change in X so the slope of the tangent line here the slope is equal to our change in Y negative four over our change in X and this is just this is going to be equal to negative two so this is equal to negative two so this simplifies to f prime of 4 is equal to I'll just in a new color three squared is nine times three is 27 times negative two times negative two which is equal to negative 54 so f prime of 4 is negative 54