 # How do I differentiate a simple polynomial?

## 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:

Polynomial: $$x^2+8x+13$$

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)

Polynomial: $$3x^2 + x + 9$$

Derivative: $$6x + 1$$

Polynomial: $$4x^4 + 3x^3 + x + 19$$

Derivative: $$16x^3 + 9x^2 + 1 + 0$$

That's all there is to taking the derivative of a polynomial!