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What Is Implicit Differentiation?
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Feb 19, 2025
Differentiation is a process of finding the rate of change of a function with respect to one of its variables. In other words, differentiation gives us information about how a function changes as its inputs change. There are many different ways to differentiate a function, but one of the most commonly used methods is called implicit differentiation. Implicit differentiation is a method of differentiating a function that is not given in explicit form (i.e. in terms of x and y). In this video, we'll take a look at what implicit differentiation is and how it can be used to find the derivative of a function.
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0:01
so thus far we've been talking about
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finding derivative of functions that are
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expressed excuse me in the explicit form
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I'm going to give you an example
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for example if I have y equals
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um
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three
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x squared minus 5. this is called the X
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explicit
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form of this function of the function
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okay
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explicit and why do we use the term
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explicit because the function is clearly
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defined you know that Y is in terms of X
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Y equals 3x squared minus 5. now to find
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the derivative of this this is pretty
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simple if you want to find the
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derivative of this function in terms of
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X we call that d y over DX right or you
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can call that y Prime that's another way
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too
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um to write down derivative it's just
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going to be straightforward okay we use
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the power rule and that's going to give
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you three times two times x to minus 1
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and the derivative of 5 or any constant
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is zero so this ends up being 6 x okay
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so that's the derivative so now the
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problem comes when sometimes the
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function is not expressed expressed
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explicitly in the sense that it is not Y
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is not clearly defined it is implied let
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me give you a um an example if I have x
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squared
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find his wife with minus y squared plus
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three y cubed equals to five and the
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question is find the derivative now as
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you can see here what's going on here Y
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is not clearly defined in terms of X so
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this is called the implicit form of the
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expression or the function the implicit
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something is being implied here implicit
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form now I'll give you an easier case so
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you can understand that so let's say I
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have
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x times y equals one I can clear this
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off away here right
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so y will be equal to 1 over X now this
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is called explicit
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form
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and this is called the implicit implicit
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because
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something is being implied here
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basically you can solve y in terms of X
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this is the common case right so you can
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solve y in terms of X this is called
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implicit format now this is called
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explicit because y's is clearly defined
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in terms of X now how do you find the
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derivative of a function like this
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this is more complicated now first let's
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take a couple of cases let's assume that
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we have this here
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I want to find D
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of DX of 3x squared okay I want to find
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D of DX of 3x square is called the
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derivative okay you want to
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differentiate this function this is
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pretty much using the power rule here
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because the variables are green right I
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have a x here and I have X they agree so
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variables agree because I'm doing this
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in terms of x
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so that's just going to give me three
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times two times x squared minus one
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basically this is 6 x right so this is
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called an explicit uh format this is
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clearly defined the variable agree
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D of t x of three x square now I get 6X
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now what happens is if I want to find
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the derivative
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of the function y cubed
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DX
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so now as you can see here I'm finding
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this derivative in terms of X what a
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variable on top is y y cubed so how do
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you solve this we use again the chain
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rule format okay let me recall the chain
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rule chain rule
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if you guys remember the chain rule
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so if you have any function
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you to the path n and you want to find
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this derivative okay derivative
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it gives you what n times U to the N
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minus 1 times U Prime okay that's the
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general form of a
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um when you want to use the chain rule
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we're going to apply that here now again
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the variables in terms of X so basically
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the derivative is going to give you 3
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times y squared
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which is basically three minus one right
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times d y over DX we're assuming that
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this is the derivative okay
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I'm going to give you 3y squared times y
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Prime basically this is utilizing again
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the chain Rule and why do we do that
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because the variables do not agree you
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have an X here you have a y cubed here
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so you have to use the chain rule to
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solve this problem so it's going to be 3
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times y 3 minus 1 times d y over DX
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because these two variables do not agree
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this is why and this is how we use
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and derivative and to find it implicit
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to use the implicit format okay to solve
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for derivative of a function this is
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called differentiation using the
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implicit format so now again let me
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recap this so if you have
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a function like this right where Y is
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clearly defined in terms of X you just
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go straight up and just find it as usual
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this is going to be 6X and in this case
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here Y is not in terms of X so as a
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matter of fact let's try and work on
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this problem so I hope you you wrote
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this down guys so let me do this so
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let's assume that we wanted to find the
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derivative of this function because this
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is written in the implicit format okay
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I'm finding the derivative in terms of X
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so in terms of X DX right so the
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derivative here because the variable is
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agree is just going to be 2x now we have
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a case where we have a y and we are
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trying to find the derivative so I'm
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going to use the chain rule it's going
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to be minus
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2y times what d y over DX
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and then plus again I have another case
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3 times 3 right times y squared
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d y over DX I'm using the chain Rule and
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then the derivative of the constant is
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zero so now we have two x right
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minus 2y
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d y
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over DX
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plus 9 y squared d y over DX equals to
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zero now we are not done we are solving
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for d y over DX we want to find the
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derivative in terms of X but y here is a
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it's not a constant again it's a
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variable and they do not agree so we are
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solving for d y for DX
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so from here what can we do we can put d
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y over DX as a
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um your your GCF your greatest common
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factor you're going to pull it out so d
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y over DX
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times negative 2y that's 9 squared 9y
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squared plus 2X equals to zero okay and
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we're going to shift this to the other
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side so I'm going to continue here
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so that's going to go back here so we
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have
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UI over DX times
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negative 2y
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plus 9y squared
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equals to negative 2X
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and now to solve for d y over DX I'm
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just going to divide it both sides by
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this expression so it's going to be d y
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over DX equals negative 2x over negative
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2y
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that's nine y squared and this is how
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you find you use the implicit
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differentiation to find the derivative
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of a function where Y is not clearly
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defined okay so that's it for today
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thank you
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