Taking a Derivative:
The simplest way to understand a derivative is as a formula for finding the slope of a curve. Or, as a formula that tells you the rate of change of a certain function. When you take a function’s derivative, you are finding that function that provides the slope of the first function. Fortunately, calculating a derivative is simple for some functions, although it can get more complicated as we move on.
First, we will take the derivative of a simple polynomial: \(4x^2+6x\). The first step is to take any exponent and bring it down, multiplying it times the coefficient. In other words, bring the 2 down from the top and multiply it by the 4. Then reduce the exponent by 1. The final derivative of that \(4x^2\) term is \((4*2)x^1\), or simply \(8x\).
The second term is \(6x\). Since the exponent of x is implied to be 1, we can bring that down and multiply, which does not change the coefficient. Reducing the exponent by 1 makes it 0, so the derivative of \(6x\) is just \(6x^0\), or the constant 6. Clearly, for any linear term (a coefficient times a variable with an exponent of 1) like 6x, 9x, or -2x, the derivative will simply be that coefficient.
Remember, the original polynomial we were differentiating (taking the derivative of) was \(4x^2+6x\). We can go term-by-term, so the final derivative is determined to be 8x + 6. That's it!
Now let's take the derivative of a few more polynomials to make sure we understand the basics:$$ x^2+8x+13 $$
$$ 3x^2 + x + 9 $$
Derivative = \(2x + 8\) (notice that any constant is eliminated, because 13 is the same as \(13x^0\), and when the 0 comes down the whole term becomes 0 and disappears)
$$ 4x^4 + 3x^3 + x + 19 $$
Derivative = \(6x + 1\)
Derivative = \(16x^3 + 9x^2 + 1 + 0\)
That's all there is to taking the derivative of a polynomial!