Differentiation involving summation

How would I differentiate an equation inlvoling a sum?

e.g.

$\frac{d}{dt}\sum_{i=1}^n ln(1+tx_i)$

Thank you!

How would I differentiate an equation inlvoling a sum?

e.g.

$\frac{d}{dt}\sum_{i=1}^n ln(1+tx_i)$

Thank you!

How would I differentiate an equation inlvoling a sum?

e.g.

$\frac{d}{dt}\sum_{i=1}^n ln(1+tx_i)$

Thank you!

You already know that the derivative of a sum is the sum of the derivatives, I take it?

You already know that the derivative of a sum is the sum of the derivatives, I take it?

How would you prove this. ive never come across stuff like this.is this first year undergrad or something?


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is it just u differentiate each term independantly like x+x^2


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First-year undergrad, yes. From the definition of 'derivative': $\displaystyle (f+g)'(x) = \lim_{h \to 0} \dfrac{(f+g)(x+h)-(f+g)(x)}{h} \\ = \lim_{h \to 0} \left( \dfrac{f(x+h) - f(x)}{h} +\dfrac{g(x+h)-g(x)}{h} \right) \\ = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} + \lim_{h \to 0} \dfrac{g(x+h) - g(x)}{h} = f'(x) + g'(x)$.

If it was an innfinite sum then the question would be a little bit more involving

If it was an innfinite sum then the question would be a little bit more involving

How so? I'm intrigued.

There is a theorem which states that all power series have "circles of convergence": if they converge for $z \in \mathbb{C}$ then they converge for all $w \in \mathbb{C}$ with $|w| < |z|$.

There is a further theorem which states that all power series are differentiable inside their circles of convergence, and the derivative is the termwise derivative of the power series. (It's a several-part theorem: the termwise derivative of the power series does converge, the power series is differentiable, and the termwise derivative converges to the derivative of the power series.) This is a bit involved to prove: I've seen two proofs, one of which seemed pretty un-motivated to me, and one which goes by the notion of uniform continuity. It's not at all an obvious result, though.

Well, I suppose it's an improvement that I understood most of those words.

Seriously though, I've heard of that before, this should really motivate me into doing some sequences and series. Can't find any resources for them though.

There is a theorem which states that all power series have "circles of convergence": if they converge for $z \in \mathbb{C}$ then they converge for all $w \in \mathbb{C}$ with $|w| < |z|$.

There is a further theorem which states that all power series are differentiable inside their circles of convergence, and the derivative is the termwise derivative of the power series. (It's a several-part theorem: the termwise derivative of the power series does converge, the power series is differentiable, and the termwise derivative converges to the derivative of the power series.) This is a bit involved to prove: I've seen two proofs, one of which seemed pretty un-motivated to me, and one which goes by the notion of uniform continuity. It's not at all an obvious result, though.

What level of knowledge are you at?

The result quoted by Smaug is a fairly standard piece of complex variable theory, probably covered in 2nd year at a 'decent' university.

There is a real-variable equivalent which would be covered as part of 1st year Analysis, so you should be able to find some good resources in any half-decent Analysis textbook (either real or complex),

*coughs* Still self-studying A-Levels. I'm comfortable with most of C1-4, FP1 and M1. So yeah, this is a little out of my depth, but I rather like being out of my depth.

how old are u?


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First-year undergrad, yes. From the definition of 'derivative': $\displaystyle (f+g)'(x) = \lim_{h \to 0} \dfrac{(f+g)(x+h)-(f+g)(x)}{h} \\ = \lim_{h \to 0} \left( \dfrac{f(x+h) - f(x)}{h} +\dfrac{g(x+h)-g(x)}{h} \right) \\ = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} + \lim_{h \to 0} \dfrac{g(x+h) - g(x)}{h} = f'(x) + g'(x)$.

I'm not an undergrad, but wouldn't a simpler proof be a geometric one? The sum of the functions' gradients will be the same as the gradient of the whole function due to the individual functions each contributing to the whole function so the rates of change MUST clearly be equal. Isn't it intuitively obvious? Or are there some cases I've not considered here.