What is Chain rule: Definition and 508 Discussions
In calculus, the chain rule is a formula to compute the derivative of a composite function. That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite f ∘ g — the function which maps x to
f
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{\displaystyle f(g(x))}
— in terms of the derivatives of f and g and the product of functions as follows:
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{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}
Alternatively, by letting h = f ∘ g (equiv., h(x) = f(g(x)) for all x), one can also write the chain rule in Lagrange's notation, as follows:
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{\displaystyle h'(x)=f'(g(x))g'(x).}
The chain rule may also be rewritten in Leibniz's notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x (i.e., y and z are dependent variables), then z, via the intermediate variable of y, depends on x as well. In which case, the chain rule states that:
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{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}
More precisely, to indicate the point each derivative is evaluated at,
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{\displaystyle \left.{\frac {dz}{dx}}\right_{x}=\left.{\frac {dz}{dy}}\right_{y(x)}\cdot \left.{\frac {dy}{dx}}\right_{x}}
.
The versions of the chain rule in the Lagrange and the Leibniz notation are equivalent, in the sense that if
z
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{\displaystyle z=f(y)}
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{\displaystyle y=g(x)}
, so that
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{\displaystyle z=f(g(x))=(f\circ g)(x)}
, then
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{\displaystyle \left.{\frac {dz}{dx}}\right_{x}=(f\circ g)'(x)}
and
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{\displaystyle \left.{\frac {dz}{dy}}\right_{y(x)}\cdot \left.{\frac {dy}{dx}}\right_{x}=f'(y(x))g'(x)=f'(g(x))g'(x).}
Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x. As put by George F. Simmons: "if a car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man."In integration, the counterpart to the chain rule is the substitution rule.
Hi, I'm keep studying The Road to Reality book from R. Penrose.
In section 12.4 he asks to give a proof, by use of the chain rule, that the scalar product ##\alpha \cdot \xi=\alpha_1 \xi^1 + \alpha_2 \xi^2 + \dots \alpha_n \xi^n## is consistent with ##df \cdot \xi## in the particular case...
I'm having trouble(s) showing that unit polar bases do not commute.
Adapting <https://math.stackexchange.com/questions/3288981>
taking: ##\hat{\theta} = \frac{1}{r}\frac{\partial }{\partial \theta} ( =\frac{1}{r}\overrightarrow{e}_{\theta})##
then ##\hat{r}\hat{\theta} = \frac{\partial...
My interest is only on the highlighted part, i can clearly see that they made use of chain rule i.e
by letting ##u=1+x^2## we shall have ##du=2x dx## from there the integration bit and working to solution is straightforward. I always look at such questions as being 'convenient' questions.
Now...
For part(a),
The solution is,
However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)##
Many thanks!
This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example
## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ##
and not
## 2 \bf{v} \cdot \nabla \bf{v} ##
Is there an obvious way to see / think of this...
I want to follow the LienardWiechert potential derivation in Robert Wald's EM book, page 179. I do not understand $$dX(t_\text{ret})/dt$$ on the right side. I assume the chain rule is applied, but I can't see how.
$$ \frac{\partial[x'^i  X^i(t  \mathbf x  \mathbf x'/c)]}{\partial x'^j} =...
I am currently selfstudying Taylor and Mann's Advanced Calculus (3rd edition, specifically). I stumbled across their guidelines for a proof of the chain rule, leaving the rest of the proof up to the reader to complete.
I was wondering if someone could look over my proof, and point out any...
Hi everyone
In the below problem, I understand that the chain rule is being used. The derivative is then equated to zero. Since the derivative is composed of dy/du and du/dx, the derivative will equal zero if either dy/du or du/dx equals zero.
However, u would be everything under the square...
I'm having some problems using the chain rule and I'm not sure where the trouble lies. For example:
If I'm not mistaken, if we have the composite function f(g(n)) then \Delta f(g(n)) = \dfrac{ \Delta f(g) }{ \Delta g } \dfrac{ \Delta g(n) }{ \Delta n }
Let f(g(n)) = (n^2)^2. Then f(g) = g^2...
I originally thought you’d have to use the chain rule to get h’, as in: f’(g(x))*g’(x). Plugging in 1 for x, I got an answer of 10. An online solution, however, said that you only had to get f(g(1)), which was f(1), then look up f’(1) in the table. Both approaches seem logical to me, but they...
In Chapter 3 of Thomas’s Calculus, they give the following proof of the Chain Rule. After the proof, the text says that this proof doesn’t apply when the function g(x) oscillates rapidly near the origin and therefore leads delta u to be 0 even when delta x is not equal to 0. Doesn’t this proof...
If Tl;dr I am struggling in Math 171 and Physics 191 and throwing around the idea of declaring a geology major with an astronomy minor because the Physics major "juice is not worth the squeeze" at my age(29) anyone else out there who struggled with Calculus 1 when they first took it?Hello...
Hi,
I was attempting the following problem and am a bit confused because I don't quite understand the structure/notation of how mutual information and entropy work when we have conditionals (ab) and multiple events (a, b).
Question: Prove the chain rule for mutual information.
I(X_1, X_2...
##f_x=3*x^2+y##
##f_y=2*y+x##
##(3*(t^2)^2+e^{t1})*2*t+(2*e^{t1}+t^2)*e^{t1}##
Well, I am not sure how to evaluate it.
I got a wrong result by multiplying by 0.1, i.e.
##((3*(t^2)^2+e^{t1})*2*t+(2*e^{t1}+t^2)*e^{t1})*0.1##
I guess it is trivial but I am lost. :(
But, If I use chain rule than, I get that.
##\vec v_i = \frac{dr_i}{dt}=\sum_k \frac{\partial r_i}{\partial q_k} \cdot \frac{\partial q_k}{\partial t}## But, they found that?
Hi, PF
##y^2=x## is not a function, but it is possible to obtain the slope at any point ##(x,y)## of the equation without previously clearing ##y^2##. It's enough to differentiate respect to ##x## the two members, treat ##y## like a ##x## differentiable function and make use of the Chain Rule...
So first thing I tried was to separate the variables then differentiate by parts, setting u = E and v = 1/ln(E) (and the other way around) but I couldn't do the integral it gave.
Then I tried to reason that because dx was constants then dE/dx is equal to E/x but I was told that's not the case...
First I quote the text, and then the attempts to solve the doubts:
"Proof of the Chain Rule
Be ##f## a differentiable function at the point ##u=g(x)##, with ##g## a differentiable function at ##x##. Be the function ##E(k)## described this way:
$$E(0)=0$$...
I have no answer or solution to this. So I'm trying to seek a confirmation of whether this is correct or not:
##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial t}dt ##
##\frac{df}{dt} = \frac{\partial f}{\partial x} \dot x + \frac{\partial f}{\partial t} ##
Therefore,
##...
I want to take the derivative of a composite function that looks like
$$f( g(x), h(x) ).$$
I know from Wolfram that the answer is
$$\frac{ df( g(x), h(x) ) }{ dx } = \frac{ dg(x) }{ dx }\frac{ df( g(x), h(x) ) }{ dg(x) } + \frac{ dh(x) }{ dx }\frac{ df( g(x), h(x) ) }{ dh(x) }.$$
We can...
Hello! Now this is not really a physics problem of the usual kind but I'd say you could consider it one.Still I'd like to post my problem here because here I always get great help and advice.Now for this problem in particular,it is in the section of the book that deals with derivatives so I...
I'm coming back to maths (calculus of variations) after a long hiatus, and am a little rusty. I can't remember how to do the following derivative:
##
\frac{d}{d\epsilon}\left(\sqrt{1 + (y' + \epsilon g')^2}\right)
##
where ##y, g## are functions of ##x##
I know I should substitute say ##u = 1...
The following link leads to a question I asked on the mathematics Stack Exchange site.
https://math.stackexchange.com/questions/3790900/chainrulewithafunctiondependingonfunctionsofdifferentvariables/3791017?noredirect=1#comment7809514_3791017
I want to understand how the chain rule...
Mentor note: Fixed the LaTeX in the following
I have the following statement:
\begin{cases} u=x \cos \theta  y\sin \theta \\ v=x\sin \theta + y\cos \theta \end{cases}
I wan't to calculate:
$$\dfrac{\partial^2}{\partial x^2}$$
My solution for ##\dfrac{\partial^2}{\partial...
I literally don't know what's going wrong today, I can't seem to get anything right :oldconfused:. The velocity in S' is easy enough $$v' = \frac{dx'}{dt'} = \frac{\partial f}{\partial t} \frac{\partial t}{\partial t'} + \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}\frac{\partial...
find $F'(x)$
$$F(x)=(7x^6+8x^3)^4$$
chain rule
$$4(7x^6+8x^3)^3(42x^5+24x^2)$$
factor
$$4x^3(7x^3+8)^3 6x^2(7x^3+4)$$
ok WA returned this but don't see where the 11 came from
$$24 x^{11} (7 x^3 + 4) (7 x^3 + 8)^3$$
I am looking at the derivation for the Entropy equation for a Newtonian Fluid with Fourier Conduction law. At some point in the derivation I see
\frac{1}{T} \nabla \cdot (\kappa \nabla T) =  \nabla \cdot (\frac{\kappa \nabla T}{T})  \frac{\kappa}{T^2}(\nabla T)^2
K is a constant and T...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some further help in order to fully understand the proof of Theorem 8.15 ...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some help in order to fully understand the proof of Theorem 8.15 ...
Theorem 8.15...
y(x,t) = 1/2 h(xvt) + 1/2 h(x+vt)
This is from the textbook "quantum mechancs" by Rae.
The derivative is given as dy/dt = v1/2 h(xvt) + v1/2 h(x+vt)
I'm not quite sure how this is? If I use the chain rule and set the function h(xvt) = u
Then by dy/dt = dy/du x du/dt I will get (for the...
MTW p 257, exercises 10.2 through 10.5: These exercises are all dealing with this familiar property of derivatives ∇ (AB) = ∇A B + A ∇ B . I learned this was called the "product rule". I learned that d/dx f(y(x)) = df/dy dy/dx is called the "chain rule". MTW keeps calling what I learned as the...
prove that if ##g:Y→Z## and ##f:X→Y## are two smooth maps between a smooth manifolds, then a homomorphism that induced are fulfilling :## (g◦f)∗=f∗◦g∗\, :\, H∙(Z)→H∙(X)##
I must to prove this by a differential forms, but I do not how I can use them .
I began in this way:
if f∗ : H(Y)→H(X), g∗...
Summary: Failed find information on the internet, really appreciate any help.
Can someone tell me what is ∇ϒ∇δ𝒆β? It seems to be equal to 𝒆μΓμβδ,ϒ+(𝒆νΓνμϒ)Γμβδ. Is this some sort of chain rule or is it by any means called anything?
In D Alembert's soln to wave equation two new variables are defined
##\xi## = x  vt
##\eta## = x + vt
x is therefore a function of ##\xi## , ##\eta## , v and t.
For fixed phase speed, v and given instant of time, x is a function of ##\xi## and ##\eta##.
Therefore partial derivative of x w.r.t...
For context, we have an equation f(x,y) = \frac{x}{y} and we had used the substitutions x = r \cos\theta and y = r \sin\theta . In the previous parts of the question, we have shown the following result:
\frac{\partial f}{\partial x} = \cos\theta \Big(\frac{\partial f}{\partial r}\Big) ...
I had already calculated the first partial derivative to equal the following:
$$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}$$
Now the second partial derivative I can use the chain rule to do and get to...
If we have an equation ##g (q,w) =f(q,w)## and we want to find the derivative of that equation with respect to w, we would normally do $$\frac {dg}{dw} = \frac {d}{dw} f(q,w) = \frac {df}{d(w)} \frac {d(w)}{dw} = \frac {df}{d(w)} $$ but my friend is saying that $$\frac {dg}{dw}= \frac...
Homework Statement
https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1007&context=foundation_wave
I'm trying to understand this paper and I'm stuck at equation (7.8). That part of the text is very short so I hope you don't mind if I don't copy the equations here.
Homework...
Hello Forum,
When the force ##F## and its resulting acceleration ##a## have the most general form, the acceleration can depend on the position ##x##, time ##t## and speed ##v##. Newton's second law is given by ## \frac {d^2x}{d^2t}= a(x,t,v)##.
When the acceleration is only a function of...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...
I need help in order to fully understand Theorem 12.7, Section 12.9 ...
Theorem 12.7...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...I need help in order to fully understand Theorem 12.7, Section 12.9 ...Theorem 12.7...
Hello,
I'm trying to follow Goldstein textbook to show that the Lagrangian is invariant under coordinate transformation. I got confused by the step below
So
## L = L(q_{i}(s_{j},\dot s_{j},t),\dot q_{i}(s_{j},\dot s_{j},t),t)##
The book shows that ##\dot q_{i} = \frac {\partial...
find y'
$$y=\sqrt{7x+\sqrt{7x+\sqrt{7x}}}$$
ok this was on mml but they gave an very long process to solve it
don't see any way to expand it except recycle it via chain rule
any suggest...