# 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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### Given $\theta = \arccos(5x^3)$, finding $d\theta/dx$ implicitly and with the chain rule

I am having trouble with a question on my homework and I have done some work but I'm not sure if I have completed all that is asked or how to take it further if needed. Let $\theta = \arccos(5x^3)$. ...
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### Using chain rule to find a partial derivative

I'm currently studying for the math GRE and this question was on one of the recent practice exams. I will restate the full question below, then provide my solution. Let $u(x,y)$ and $v(x,y)$ be real-...
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### Correct Application of Chain Rule?

\begin{align*} &\frac{d}{dx} 4(2x + \sqrt{x^3+3})^{-2}\\\\ =\text{ }&4 \cdot \frac{d}{dx} (2x + \sqrt{x^3+3})^{-2}\\\\ =\text{ }&4\frac{-2}{(2x+\sqrt{x^3+3})^3} \cdot \frac{d}{dx}(2x + \...
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### Does $y\to x,y’’\to-\frac{y’’}{y’^3}$ always convert a differential equation into that of the inverse function’s?

The inverse substitution $y\to x$ implies $\frac{dy^{-1}(x)}{dy}=\frac1{y’(x)},\frac{d^2y^{-1}(x)}{dy^2}=-\frac{y’’(x)}{y’^3(x)}$, so $y’\to \frac1{y’}$ and $y’’\to-\frac{y’’}{y’^3}$. It seems to ...
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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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### Chain rule for integrator function with Riemann-Stieltjes integral

I have a random variable $X$ with distribution function $F$. I am interested in evaluating the integral $$\int g(x) \beta'(F(x)) F(dx)$$ where $\beta$ is smooth and monotone and $g$ is such that the ...
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### Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$
Say I have a smooth vector-valued function $h:\mathbb{R}^n\to\mathbb{R}^n$ and a smooth diffeomorphism $\varphi:\mathbb{R}^n\to\mathbb{R}^n$. Consider the gradient of the composition \$h\circ\varphi^{-...