SECTION 3.1: Derivative Formulas for Powers and Polynomials

The derivative of constant is 0 as the slope of the line passing through that point is also 0.
Hence is f(x) = k, it implies that f ' (x) = 0.

The derivative of a linear function refers to the slope of that line, i.e.,
if f(x) = mx +b, f ' (x) = m

A function can be multiplied by a constant. If this is done, the general shape of the graph changes depending on the magnitude of the constant and its sign. The magnitude changes the slope of the curve at each point while the sign determines whether the function is reflected over the x-axis or not (reflected only if negative).
If c is a constant, then d/dx [c*f(x)] = c*f ' (x)

Derivatives of sums and differences of two functions can be found by simply adding or subtracting their magnitudes, i.e.,
d/dx [f(x) + g(x)] = f ' (x) + g ' (x) and,
d/dx [f(x) - g(x)] = f ' (x) - g ' (x)

Theoretically and graphically proven rules of differentiation:

The power rule:
d/dx (x^n) = n*[x^(n-1)]

The constant multiple rule:
d/dx [3*(x^5)] = 3*[d/dx(x^5)] = 3*[5*(t^4)] = 15(t^4)

The sum rule:
d/dx [(p^5)+(p^3)] = d/dx (p^5) + d/dx (p^3)

Similar rules can be used when differentiating to find the second derivative of a function. Finding the second derivative implies finding the derivative of the derivative of a function.

It is confusing to understand exactly what the graph of a second derivative aims to show of a function. It also brings up the question of whether a third or fourth derivative can also exist in a real world situation.

These formulae can be useful in determining general trends of markets. Anything related to graphs and relations of functions such as this can be useful in the field of economics.