To find the derivative of a polynomial, apply the same pattern to each term of the polynomial. If a term of the polynomial is aX^n, the derivative of that term is anX^(n - 1). For the constant term -- where there is no X -- it should be noted that the derivative of a constant is zero. This makes sense, because the way a constant changes when X changes is "not at all" or zero. For example, if Y = X^3 + 3X^2 - 3X -7, then dY/dX = 3X^2 + 6X - 3.
The derivative of a function represents the slope of the tangent line to the curve at any point. If Y = X^3 + 3X^2 - 3X -7, the derivative dY/dX = 3Z^2 + 6X - 3 gives the slope of the tangent lines. At the point where X - -3, the slope of the tangent line is 3(-3)^2 + 6(-3) - 3 = 6. The tangent point is where X = -3 and Y = (-3)^3 + 3(-3)^2 - 3(-3) - 7 = 2. The tangent line has slope 6 and goes through the point (-3, 2). From basic algebra this means the line Y = 6X + 20 is tangent to Y = X^3 + 3X^2 - 3X -7 at (-3, 2)
The tangent line is horizontal when the curve changes direction. This point represents a local extrema -- minimum or maximum. Calculus gives us a way to find extrema of a function. Set the derivative to zero, and solve this equation to find the extrema. For example, if 100 feet of fence is used to enclose an area next to a barn, the area is A = L(100 - 2L), where L is the length of fence perpendicular to the barn. A = 100L - 2L^2 and dA/dL = 100 - 4L. Setting dA/dL = 0 = 100 -4L means that the maximum area will be enclosed when 25 feet of fence is perpendicular to the barn.
The most valuable use for derivatives is to describe natural phenomena. The derivative of acceleration is velocity and the derivative of velocity is distance -- velocity is measuring how fast distance is changing. If you are riding in the back of a pickup truck with a piano, you will see an example of the descriptive power of derivatives. The piano moves when the truck is accelerating or decelerating. At maximum speed -- or when the truck is stopped -- the piano does not move. The movement of the piano is the derivative of the movement of the truck.