This is a bit tricky to answer because the notion of 'rate of change' is to me the easy explanation of what the derivative is.
Let's start with what do you know?
In particular how does any of the following help you?
Differentiation from first principles.
Suppose we want to know the slope of the graph of some function at a point. We can try drawing the tangent line by hand and hoping we get a good enough fit and then finding its gradient (it's just a straight line after all). But the problem there is the standard 'real life does not correspond to the perfect world we imagine mathematically'.
Instead let's think about drawing a little chord from the point on the curveto a little point a bit further along, say e in the x direction. The smaller we let e get the better that chord's slope approximates the tangent's.
In numbers, if f is the function, we want to know what happens in
{f(x+e)-f(x)}/e
as e gets small.
To see where that expression comes from, try drawing the graph of something, and picking some x, some e and 'joining up the dots... '
I'm sorry, anyone know if the tex mode here allows us to use xypic?
In words, that quantity is looking at the instantaneous rate of change if you like. We'll try and explain why later.
Let's do an example
x^n
Work out the binomial expanion of (x+e)^n and subtract x^n, divide everythin by e and what happens as e gets small? As is often the case, here you can just set e to be zero and nothing goes wrong. You should get nx^(n-1).
Do the same for sin, knowing that for e small sin(e) is practically e.
Let's try and get back to rates of change.
Kinetics. Suppose we are doing th standard equations of motion with a fixed acceleration.
suppose our initial velocity is u, we accelerate for e seconds what's our final velocity? u+ae where a is the acceleration.
Now, a is the rate of change of speed with respect to time, as we all know and understand. What this tells us is, writing it more formally,
let v(t) be the velocity at time t
v(t+e)=v(t) + e.a
where a can be thought of as dv/dt
generally, dv/dt isn't constant, and the above equation should be approximate. So another way to interpret (actually just the same but rewritten) derivatives is
f'(x) is the function that makes
f(x+e) apporximately equal to f(x) + e.f'(x)
it's the linearization of the error if you like. The approximate we needn't worry about too much at the moment.
How much of that do you know?
What do you mean by 'solve' an application? Perhaps if you posted an example of what you were trying to solve?
