Are the differential and derivative of a single-variable function exactly the same thing? I just started taking a calculus class but I got in late and it had already started like weeks ago, so I'm completely lost.
I believe the teacher uses this same formula in order to get the differential and derivative of a function
$$\lim_{h\to0}\frac{f(x + h) - f(x)}{h}$$
Are they the same? I just need some help to understand it because I can calculate it using this formula (in case it's possible to remove the $h$ from the denominator) but I have no idea of what are these for, I need some examples of when am I going to need them.
 A: I assume that you know what is the meaning of being differentiable at a point $x$. Let the function $f$ is differentiable at $x$ ($f'(x)<\infty$). Then we can have-as that lmit tells- the following identity: $$\Delta f=f(x+\Delta x)-f(x)=f'(x)\Delta x+\epsilon\Delta$$ such that this $\epsilon$ is a function with respect to $\Delta x$ and of course $\epsilon\to 0$ when $\Delta x\to 0$. The part $f'(x)\Delta x$ is differential of the function $f$ itself (at $x$). For small $\Delta x$ $df=f'dx$ is a good approximation to $\Delta f$. For example if $y=\sin x$, then $\Delta y=\sin(x+\Delta x)-\sin(x)$ and $dy=\cos(x)\Delta x$ and so $\sin(x+\Delta x)\approx\sin(x)+\cos(x)\Delta x$.

A: You have written the definition of the derivative of a function $f(x)$ in that formula.
A differential is an infinitesimal interval; $dx$ for example.  We use differentials in expressions like $\dfrac{dy}{dx}$.  As a new calculus student, you want to focus on finding derivatives of functions.  A derivative is a rate of change, just like the slope of a line, but with derivatives you can find the rate of change of non-linear functions.
You use the limit definition above to find the derivative of functions.  As you progress in your understanding of calculus, you will memorize patterns instead of using that definition every time.
Don't worry about the term differential too much for now.  Do learn how to take derivatives of functions.
