# 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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### How does the square root disappear when differentiating $y=\frac{\sqrt{2x^2}}{\cos x}$?

Finding the derivative of $$y=\frac{\sqrt{2x^2}}{\cos x}$$ I am going through the steps and having trouble using the quotient rule. I have seen the final answer, and I've had no trouble using the ...
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### Chain rule for subdifferentials of nonconvex functions

I have two functions: one of them $h\colon\mathbb{R}^n\to\mathcal{S}$ is smooth, but not necessarily convex, and the other $g\colon\mathcal{S}\to\mathcal{S}$ is convex, non-expansive, and not ...
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### Clarifying the chain rule terminology in differential geometry calculuations

Let $M$ be a manifold and $f:M\to\mathbb{R}$ a smooth function on it. Let $p\in M$ have the coordinates $\{x^i\}$ under the chart $(U,\phi)$. Finally, let $\gamma:I\to M$ be a curve ($I$ is an open ...
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### If $g$ is smooth and $f$ is convex non-smooth, then $\partial f(g(x))\cdot Dg(x)\subseteq \partial (f\circ g)(x))$?

Let $g\colon\mathbb{R}^n\to\mathbb{E}$ be a smooth function and $f\colon\mathbb{E}\to\mathbb{E}$ be a convex function (for my purposes, a projection onto a convex set), where $\mathbb{E}$ is an $m$-...
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### The temperature of a metal plate is given by $f(x,y)=\frac{150}{\sqrt{x^2+y^2+1}}$. Find the R.O.C of the temperature at the point $(8,4)$…

The temperature of a metal plate is given by $f(x,y)=\frac{150}{\sqrt{x^2+y^2+1}}$. Find the rate of change of the temperature at the point $(8,4)$, in the direction towards the point $(7,2)$ Right ...
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