Yes.
y = e^x
dy/dx = e^x
Note: e^x = 1 + x + x²/2! + x³/3! + ... + x^r/r! + ...
→ d/dx(e^x) = 1 + x + x²/2! + x³/3! + ... + x^(r-1)/(r-1)! + ...
(The font is extremely bad: r! is r-exclamation mark = r factorial = r × (r-1) × (r-2) × ... × 2 × 1)
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It is a function. If the graph contains at least two points on the same vertical line, then it is not a function. This is called the vertical line test.
it is the same as a sin function only shifted to the left pi/2 units
To shift a funcion (or its graph) down "a" units, you subtract "a" from the function. For example, x squared gives you a certain graph; "x squared minus a" will give you the same graph, but shifted down "a" units. Similarly, you can shift a graph upwards "a" units, by adding "a" to the function.
This would be graphed the same way as any other function, except that any values which are normally drawn below the x-axis are instead reflected around it.
if the question is why is it labelled as f(x) ? it means the function (the 'f') at a certain x value. saying f(x) is said as 'f at x'. it's the same as saying 'function at x'