The Chain Rule: Explanation with Examples!

up to this point you should have learned how to take the derivatives of various types of functions including polom products natural logs and more but how would we go about taking the derivative of something like this where one function is nested inside of another in this video I'm going to show you how to apply the chain rule for problems such as [Music] these before we get started let's briefly review composition of functions composition of functions can be described as nesting one function inside another and is usually denoted with a little circle between the two functions for example if f ofx = x cubed and G ofx = 8x - 2 then we would say that F composed with g equals the composite function f of g ofx equals the quantity 8x - 2 cubed to compose one function with another it's helpful to write them in nested form like this F of G ofx from here we can replace G ofx with 8x - 2 which gives us F of 8x - 2 and plug that in everywhere we see an X in the function f ofx in a similar way we could write G composed with f as G of f ofx equals g of X cubed which is 8 * X Cub - 2 the chain rule is a method which helps us take the derivative of nested functions like f of G ofx it states that the derivative of a composite function like f composed with G is equal to the derivative of the outer function composed with the inner function and this is multiplied by the derivative of the inner function another way of expressing the chain rule is by writing the derivative of the function f composed with G is equal to fime of G ofx * G Prime of X so if we wanted to find the derivative of f of G ofx we would first write the derivative of the outer function the cube while leaving the inner function alone given that f ofx = x cubed then fime of x = 3 * x^2 and frime of G of x = 3 * the quantity 8x - 2^ 2 then multiply this by the derivative of the inner function 8x - 2 which would be 8 as a result the derivative of the function f composed with G is equal to 3 * the quantity 8x - 2^ 2ar * 8 this can simplify to 24 * the quantity 8x - 2^ 2 now we could have handled this derivative by expanding out the quantity 8x - 2 cubed and then applying the power rule to the resulting polom however this expansion would have been timec consuming to calculate calate once you get the hang of the chain rule you'll find that it's an excellent timesaver in problems like this let's work through another example use the chain rule to find the derivative of the function H ofx equal the square root of the natural log of x first let's identify what the outer and inner functions are since the natural log of x is nested inside the radical it is the inner function while the outer function is the square root the chain rule says that the derivative of a composite function is equal to the derivative of the outer function with the inner function untouched times the derivative of the inner function since the derivative of the square root of x or x to the 1/2 power is 1 / 2 * x to the - 1/2 power we write the first part of the derivative as 1 / 2 * the quantity of the natural log of x to the - one2 power then multiply by the derivative of the inside since the derivative of the natural log of x is 1 /x we will multiply by 1 /x simplifying we get the H Prime of x = 1 / 2x * the quantity of the natural log of x to the -2 or 1/ the quantity 2x * the < TK of the natural log of x let's do another example use the chain rule to find the derivative of the function J of x = e raised to the quantity 8 * x^2 this function may look a little intimidating but we have the tools necessary to work through its derivative in this case the outer function is the exponential function e raised to a power the inner function is that Power 8 * x^2 to get the derivative with the chain rule we need to First write the derivative of the outer function with the inner function untouched since the derivative of the outer function e to the x is itself we are just going to write e to the quantity 8 * x^2 again then we need to multiply this by the derivative of the inner function since the derivative of 8 * x^2 is 16x let's go ahead and write that down this simplifies to 16x * e toty 8 * x^2 not so bad right now I want you to try a problem on your own using the chain rule find the derivative of K of X = the quantity X Cub - 9 to the 11th power remember you'll need to handle the outer functions derivative first then the inner functions derivative pause the video to work out the answer for yourself let's check your work what is the outer function in this problem it's the 11th power the inner function is X Cub - 9 the first part of the chain rule dictates that we write the derivative of the outer function with the inner function still inside of it so we will have 11 * the quantity X Cub - 9 to the 10th power then we are going to multiply this by the derivative of the inner function which will be 3 * x^2 if we simplify this we'll get K Prime of X is 33 x^2 * the quantity X Cub - 9 to the 10 power this problem like the first one in this video could have been solved by expanding the polom to a 12 term expression and then handling each term individually with the power rule but I think it's safe to say that the chain rule is a much better method for finding this derivative let's work through one last example Leo is studying population growth in a Petri dish of mold spores the dish starts out with only one Spore but the Spore duplicates itself and there is exponential population growth over time if the number of mold spores on day X can be calculated using the function m ofx = e to theare < TK of X determine the function M Prime of X which represents the growth rate because we are interested in finding a rate function M Prime of X we understand that this is a derivative problem and since the function m ofx is a composite function we know that we are going to need the chain rule to find that derivative the outer function in this case is the exponential function e raised to a power that power the square root of x is the inner function to find M Prime we start by taking the derivative of the outer function while leaving the inner function as is once again the derivative of e to the x is itself so we will start by just writing e to theare < TK of X again next we have to multiply this by the derivative of the inner function since the derivative of the sare < TK of X is 1 / 2 * X 1 12 power the growth rate function M Prime is equal to e to the < TK x * the quantity 1 / 2 * x^ the -2 power or equivalently E to < TK X over the quantity 2 * the < TK of X remember the chain rule helps us take the derivatives of composite or nested functions to get these derivatives first write the derivative of the outer function with the inner function still untouched within it then multiply this by the derivative of the inner function this technique will come in handy anytime you see a function contained in another function whether under a radical in a logarithm or even inside a trig function as always a little practice goes a long way so I encourage you to try some problems on your own thanks for watching and happy studying [Music]