1
vote
2answers
206 views

What is the mistake in this proof of product rule of differentiation?

I was trying to derive the product rule of differentiation which states: If $y=u\cdot v$, then, $y'=u'\cdot v+v'\cdot u$. So I assumed it like this: $y=u+u+u+\cdots$ ($v$ number of terms of $u$) ...
0
votes
2answers
27 views

Derivative: Which rule to use first?

$f(x)=x^7(5+8x)^3$ Would I go about finding the derivative of this problem by using the chain rule first, and then the product rule? Or would I do the opposite? Step by step instructions would be ...
0
votes
1answer
78 views

Proving the formula for the directional derivatives of the of the sum and dot product of two functions

Define the directional derivative of a function $\textbf{f}$ at $\textbf{c}$ in the direction $\textbf{u}$ by $$\textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) = \lim_{h \rightarrow 0} ...
2
votes
2answers
49 views

Generalizing the Product Rule

How would I go about generalizing the product rule to the product of $n$ functions $\psi_1(x), \ \psi_2(x), ..., \ \psi_n(x)$? That is, I'm hoping to obtain an expression for $$ \frac{d}{dx} \prod_{j ...
2
votes
0answers
57 views

Why does $\frac{d}{d\theta} \theta\prod_{i=1}^nx_i = \sum_{i=1}^nx_i$

Is this just the product rule? I have this in my notes but I didn't think anything of it and now I'm wondering how this happens? Edit: Im working with maximum likelihood estimation and in my notes I ...
-2
votes
4answers
254 views

Derivative of product notation?

Presume $f(x,y)$ is a continuous function. How would I take the derivative of $$\prod_{x=1}^N f(x,y)$$? Edit: derivative with respect to $x$, that is.
1
vote
2answers
107 views

Partial derivates of product

How to derive from this formula: $$\frac{\partial(\mathbf g.\mathbf h)}{\partial \mathbf x} = \left(\frac{\partial(\mathbf g.\mathbf h)}{\partial x_1},\frac{\partial(\mathbf g.\mathbf h)}{\partial ...
3
votes
1answer
94 views

Derivative of $f(s) = \frac{1}{\prod_{k=1}^{n}(x_k-s)}$

Denote $$ f'_{1}(s) = \bigg( \frac{1}{x_1-s} \bigg)'_{s} = \frac{1}{(x_1-s)^2}\\ f'_{2}(s) = \bigg( \frac{1}{(x_1-s)(x_2 -s)} \bigg)'_{s} = \frac{x_1 +x_2 - 2s}{((x_1-s)(x_2-s))^2} $$ and so on. ...