Math Calculus Derivative Knowledge Question

Q1) When you take the 1st derivative of a function. What does that tell you?

Q2) Likewise, when you take the 2nd derivative of the very same function, what does that tell you?

Q3) I know usually when after you get the 1st or 2nd derivative of the function, we have to find the x-values. What do those x-values mean?

Q4) To find the maximum and minimum, do we find the 1st derivative or the 2nd derivative?

Q5) What is the difference between relative maximum / relative minimum VS. relative extrema?

Please answer any of the above if you can. I have alot of uncertainty right now in Calculus and I just want to get it right so I don't fail. (T___T)

Q1) When you take the 1st derivative of a function. What does that tell you?


Answer: 1st derivative gives the function's rate of change. It is
the SLOPE of a tangent line to the graph of the function and can
be informally thought of as the slope of the function at the point you are
considering.

Q2) Likewise, when you take the 2nd derivative of the very same function, what does that tell you?

The second derivative actually gives the rate of change of the first derivative .

If the second derivative is zero, the first derivative is constant and you are looking at
a line . For a line, the derivative is the SLOPE of the line.

If the second derivative is GREATER than zero, then, the function is either increasing at an
increasing rate {first derivative positive} or it is decreasing at a lesser rate in absolute value.

A function whose second derivative is always greater than zero is y= x² , in fact

we could say y= ax² for any positive a.

Consider y=x² . To the right of zero, this function increases.

at x=1 the derivative y=2x is 2
at x=2 the derivative y=2x is 4
and
at x=3 the derivative y=2x is 6

Thus the derivative of the function is increasing.