# Questions tagged [chain-rule]

For questions involving the chain rule in analysis. The chain rule is a special rule to differentiate a composition (chain) of several functions. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule.

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### If $u(x) = 3x + 2y(x)$, then what is $du/dx$?

If I let $u = 3x + 2y$ where $y$ is a function of $x$, then what is $\frac{du}{dx}$? Do I use chain rule here?
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### Evaluate ${d \over da} \left({da \over db}\right)$, where $a$ is a function of $b$?

This is to help a younger sibling with their work. They didn't know how to evaluate the derivative. I want to check and make sure it was correct: Evaluate ${d \over da} \left({da \over db}\right)$, ...
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### Simple derivative computation for time dependent function

Let $F:\mathbb{R}^m\to \mathbb{R}$ be a smooth function. Consider the function $G:\mathbb{R}^m\times \mathbb{R}^m\times (0,1)\to \mathbb{R}$ defined by $$G(u,v,t)=F\left(\dfrac{u-v}{t}\right)$$ ...
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### Theorem 28.4 [Chain Rule] Ross Elementary Analysis

The following is a segment of Ross's proof of the chain rule. $f$ is defined on some open interval $J$ containing $a$. $f$ is differentiable at $a$ and suppose $f(x) = f(a)$ for $x$ arbitrarily close ...
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### Application of the Chain Rule in the Proof of Hadamard's Lemma

I read the proof for Hadamard's lemma available on Wikipedia but I do not understand the application of the chain rule in the proof. In particular, $f$ is a smooth function defined on an open star-...
1 vote
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### Chain rule when mixed partial multiplied by first order partial

Given $$r = \frac{{\partial s}}{{\partial t}}$$ If we take the partial derivative of $r$ with respect to $u$ $\frac{{\partial r}}{{\partial u}}$ and multiply it by $\frac{{\partial u}}{{\partial v}}$...
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### Sketchy use of multivariable chain rule under too weak hypoteses

I've found this statement in my real analysis course notes: Let $f: B_r (x_0) \subseteq \mathbb{R}^m \to \mathbb{R}$ ($B_r (x_0) = \{ x \in \mathbb{R}^m : d(x, x_0) < r \}$) be a function such ...
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### Multivariate Chain Rule with Derivatives in Intermediate Functions

I have a function $$G: \mathbb R^d \times \mathbb R \times \mathbb R^d \to \mathbb R$$ where $d$ is a positive integer and the arguments of $G$ are denoted by $({\bf y}, z, {\bf p})$. I'm denoting ...
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### Using the Chain Rule to Switch Between Coordinate Axes in a PDE

I have a system of PDEs that are written in terms of two coordinates: $(x,z)$ which are the usual Cartesian coordinates and $(l,n)$ which are tangential and normal components to some deformable ...
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### Why does the transformation of a random variable require the transformation have continuous derivative and the original pdf be continuous?

In Statistical Inference (2e, Casella & Berger), there is the following theorem (p. 51) (emphasis mine): Theorem 2.1.5 Let $X$ have pdf $f_X(x)$ and let $Y = g(X)$, where $g$ is a monotone ...
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### Chain rule to differentiate vector fields with a connection or Lie derivative

Let $X : M \rightarrow TM$ be a smooth vector field, $f : M \rightarrow \mathbb{R}$ a smooth real valued function and $\gamma : [0,T] \rightarrow M$ a smooth curve in $M$. I am interested in ...
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### Applying the Chain Rule for ($\sin(x) + 1$)

So, according to the chain rule, $$\frac{d(f(g(x)))}{dx} = f'(g(x)) \cdot g'(x).$$ Now, if we considered $f(x) = x+1$ and $g(x) = \sin(x)$ then: $$f(g(x)) = \sin(x)+1$$ In this case, shouldn't ...
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### Double derivative representation in terms of $\theta$?

Ok, this is a thought I had, as we all know that we can write $\mathrm dy/\mathrm dx$ as $\tan(\theta)$ because it's just the slope of the tangent line. Thought I could get the double derivative in ...
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### Application of Chain Rule in Proof of Stokes' Theorem

In the proof of Stokes' theorem in "Vector Calculus" by Marsden and Tromba, I noticed that the chain rule is applied selectively. Specifically, the chain rule is not used when expanding the ...
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### Rewriting the second derivative of a function by substitution

I would like to know if the equation $$\frac{d^2T(x)}{dx^2} = \frac{1}{2}\cdot\frac{d}{dT}\left(\frac{d}{dx}T(x)\right)^2\quad(1)$$ is true for a general function T(x). The function T(x) describes ...
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### Finding a derivative for option pricing

I am trying to find the derivative of the expression below wrt $S$: $$e^{-dt}N(x)$$ where $x$ is defined as: $$\frac{\ln({\frac{S}{K}})}{\sigma\sqrt{t}}$$ and $N(x)$ is the cumulative of the ...
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### Chain rule with a variable held constant (or along a constant surface) [duplicate]

Say we have a Cartesian function, $f(x,y)$ and we move to a polar coordinate scheme $g(r,\theta)$, $$g(r,\theta) = f(x,y) \\ \therefore x = x(r,\theta), y = y(r,\theta)$$ Just to lay out the ...
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### Strange usage of chain rule. Can anyone explain why this derivation was done this way?

There are 2 issues I have with the way this was done. The first was how chain rule was used in the (1.35), and the second was how chain rule was used in (1.36). It all seems so counterintuitive. For (...
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### The chain rule for partial derivatives

My goal is to understand a statement about partial derivatives made in the appendix of a textbook about manifolds. From the book: Let $U \subseteq \mathbb{R}^n$ and $\tilde{U}\subseteq \mathbb{R}^m$ ...
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### Chain rule and differentiability of $|x|^2$

Going through Thomas Calculus, question 90 in the chapter on Chain Rule: Suppose that $f(x) =x^{2}$ and $g(x) =|x|$. Then the composites  ( f\circ g)( x) =|x|^{2} =x^{2} \ \ \ \ \ \ and\ \ \ \ \ ( ...
1 vote
Let $z=f(u,v)$ where $u=xy$ and $v=\frac{x}{y}$ such that $f$ is second differentiable. Compute $\tfrac{{\partial}^2z}{\partial x\partial y}$ and $\tfrac{{\partial}^2z}{\partial x^2}$. My attempt for ...
### Given the wave equation, show $u_{yw}=0$
Consider the wave equation: $\alpha^2 u_{xx}=u_{tt}$. I am told to let $y=x-\alpha t$ and $w=x+\alpha t$ and use chain rule but I'm not confident in my attempt. I got $x=\frac{y+w}{2}$ and \$t=\frac{w-...