How to explain integrals and derivatives to a 10 years old kid? I have a sister that is interested in learning "what I do". I'm a 17 years old math loving person, but I don't know how to explain integrals and derivatives with some type of analogies.
I just want to explain it well so that in the future she could remember what I say to her.
 A: This is a question for a long car journey - both speed and distance travelled are available, and the relationship between the two can be explored and sampled, and later plotted and examined. And the questions as to why both speed and distance are important can be asked etc
A: Explain derivatives using the speedometer ! Ask her how one could find the speed of something , and if she goes saying the average speed , ask her how one could calculate the instantaneous speed. And boom , there comes your derivatives. 
A: Isn't there a way to describe with drawings that the derivative is the limit of the slopes of the secants, that the integral is the area under a curve, and that the derivative of the area is the height of the function at the point at which you compute derivative? Sure, she might not know what "slope" means, but certainly you can explain that to her if she has the patience ; she's not too far away from it mathematically.
If you make "clear enough" drawings, I think geometry is a good enough analogy to explain what you do. If you put in contexts where derivatives apply (physics/chemistry/economics), I think your little sister will just look at too many pictures at the same time and get lost in what you're saying. Try to be patient and stick to the "closest-to-the-real-picture portrait" of derivatives and integrals.
Oh, and one more thing ; if she isn't interested in what you do, don't lose your time. Only do it if she has the courage to listen to you, remember that mathematics are pure pain to the closed ears.
Hope that helps,
A: I quote parts of this Reddit answer.

The easiest way to understand a little calculus is to sit in the middle seat and look at the speedometer in the car.  What speed does it say?  Maybe it says 31 miles per hour.  This means that, if you keep traveling at this speed, you'll go 31 miles in an hour.  Any kid can understand that (even if the kid doesn't really know how far a mile is).  But then your dad slows down and stops at a red light.  The speed is 0 miles per hour now.  Did you actually go 31 miles in an hour?  No; 31 miles per hour was your speed only at that instant in time.  Now the speed is different.  The idea that it even makes sense to have a speed at an instant in time is... calculus!  You calculate speed by seeing how far you go and dividing by how long it took you to get there, but that only gives you average speed.  For the speed right now, you have to see how far you go in a very, very, very tiny amount of time.  You only go a very, very, very tiny distance.  And you divide by that very, very, very tiny amount of time to get a speed in numbers that you understand.  Calculus is when you make that amount of time tinier and tinier and tinier, and that makes the distance tinier and tinier and tinier too, so that, at that moment, the tiny distance divided by the tiny time is 31 miles per hour, but a second later it might be 30 mph or 32 mph or something else.
You generally use calculus to talk about how fast things change -- in the case of the car, it's how fast your position changes, but lots of things can change.  How fast something is changing right now is called the derivative.  Sometimes you know how the rate of change for something is related to other things.  For example, if you have a weight on a spring, you can write how fast the speed of the weight is changing based on its position on the spring, and you can write an equation called a differential equation.
[...]
You can also use calculus to talk about how lots of little things can add up to a big thing.  For example, let's say you have an object, and you want to know how much it weighs.  You can break it up into tiny little pieces, figure out the density for each piece, figure out how much each little piece weighs, and add them all together.  That's calculus!  (Or you can just put it on a scale -- that's physics.)
The calculus of how fast things change is called differential calculus, and the calculus of adding up lots of little things is called integral calculus.  In differential calculus, you take a tiny little number and divide by another tiny little number to get a regular-sized number.  In integral calculus, you add together a very, very, very large number of tiny little numbers to get a regular-sized number.
A: I remember the explanation I was always given.. 
"Do you know how to find the area of a circle" 
"Yes" 
"Well, do you know how to find an area of a random curved figure?" 
"No" 
"That is what we use calculus for". 
