- #1
mathmari
Gold Member
MHB
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Hey!
Let $I=[a,b]$, $J=[c,d]$ compact intervals in $\mathbb{R}$, $g,h:I\rightarrow J$ differentiable, $fI\times J\rightarrow \mathbb{R}$ continuous and partial differentiable as for the first variable with continuous partial derivative.
Let $F:I\rightarrow \mathbb{R}$.
I want to calculate the derivative of $$F(x)=\int_{g(x)}^{h(x)}f(x,y)\, dy$$ using the chain rule.
Following hint is given:
Let $G:I\rightarrow \mathbb{R}^3$ and $H:J\times J\times I\rightarrow \mathbb{R}$ defined by $G(x):=(g(x), h(x), x)=(u,v,w)$ and $H(u,v,w):=\int_u^vf(w,y)\, dy$. Then it is $F=H\circ G$. So, to calculate the derivative of the integral we have to calculate the derivative of $F(x)=H(G(x))$.
From the chain rule we have that $F'(x)=H'(G(x))\cdot G'(x)$.
The derivatives of the functions $H$ and $G$ are the following:
\begin{align*}G'(x)&=\frac{dG(x)}{dx}=\frac{dG(x)}{dg(x)}\cdot \frac{dg(x)}{dx}+\frac{dG(x)}{dh(x)}\cdot \frac{dh(x)}{dx}+\frac{dG(x)}{x}\cdot \frac{dx}{dx}\\ & =\frac{dG(x)}{dg(x)}\cdot \frac{dg(x)}{dx}+\frac{dG(x)}{dh(x)}\cdot \frac{dh(x)}{dx}+\frac{dG(x)}{x}\end{align*} The last term $\frac{dG(x)}{x}$ is not the same as $G'(x)$, is it?
For the derivative of $H$ do we use the total differential?
Or am I thinking wrong? (Wondering)
Let $I=[a,b]$, $J=[c,d]$ compact intervals in $\mathbb{R}$, $g,h:I\rightarrow J$ differentiable, $fI\times J\rightarrow \mathbb{R}$ continuous and partial differentiable as for the first variable with continuous partial derivative.
Let $F:I\rightarrow \mathbb{R}$.
I want to calculate the derivative of $$F(x)=\int_{g(x)}^{h(x)}f(x,y)\, dy$$ using the chain rule.
Following hint is given:
Let $G:I\rightarrow \mathbb{R}^3$ and $H:J\times J\times I\rightarrow \mathbb{R}$ defined by $G(x):=(g(x), h(x), x)=(u,v,w)$ and $H(u,v,w):=\int_u^vf(w,y)\, dy$. Then it is $F=H\circ G$. So, to calculate the derivative of the integral we have to calculate the derivative of $F(x)=H(G(x))$.
From the chain rule we have that $F'(x)=H'(G(x))\cdot G'(x)$.
The derivatives of the functions $H$ and $G$ are the following:
\begin{align*}G'(x)&=\frac{dG(x)}{dx}=\frac{dG(x)}{dg(x)}\cdot \frac{dg(x)}{dx}+\frac{dG(x)}{dh(x)}\cdot \frac{dh(x)}{dx}+\frac{dG(x)}{x}\cdot \frac{dx}{dx}\\ & =\frac{dG(x)}{dg(x)}\cdot \frac{dg(x)}{dx}+\frac{dG(x)}{dh(x)}\cdot \frac{dh(x)}{dx}+\frac{dG(x)}{x}\end{align*} The last term $\frac{dG(x)}{x}$ is not the same as $G'(x)$, is it?
For the derivative of $H$ do we use the total differential?
Or am I thinking wrong? (Wondering)