How do I get the derivative of a summation

Question

Sorry if this has been asked before. I'm fairly new to this. How do I find the derivative of a summation?

$$f(x)= \sum_{i=1}^{10} \frac{10x^2}i$$

what is the derivative of $f(x)$?

Answer 1

Note that $$f(x)=\sum_{i=1}^{10}\frac{10x^2}{i}$$ $$f(x)=10x^2\cdot \sum_{i=1}^{10}\frac{1}{i}$$ $$\implies f'(x)=20x\cdot \sum_{i=1}^{10}\frac{1}{i}$$ Simple derivative on $x$.

Answer 2

$$f(x)= \sum_{i=1}^{10} \frac{10x^2}i$$ $$f(x)= {10x^2}\sum_{i=1}^{10} \frac{1}i$$

Now notice that the summation is just a constant, and so your derivative is just 10 times the sum times $2x$

Answer 3

HINT: $$(f(x)+g(x))'=f'(x)+g'(x)$$ What happens if you add another function?

Answer 4

$f(x) = 10x^2/1+10x^2/2+\cdots + 10x^2/10$ Now termwise differentiate.