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Derivative of a Polynomial
Taking a Derivative:
A derivative is basically a formula for finding the slope of a curve. When you take a function’s derivative, you are finding another function that provides the slope of the first function. Finding a derivative is simple for some functions, but it can get more complicated as we move on.
First, we will take the derivative of a simple polynomial: 4x2 + 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 term is (4*2)x1, or 8x.
The second term is 6x. Since the exponent is assumed 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 6x0, or the number 6. For any linear term (a number times a variable to the first power) like 6x, the derivative will simply be that coefficient.
Remember, the original polynomial we were “differentiating” (taking the derivative of) was 4x^2 + 6x. Going term by term, the derivative was 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:
x2 + 8x + 13
Derivative = 2x + 8 (note that any constant is eliminated, because essentially that was 13x^0, and when the 0 comes down the whole term becomes 0)
3x2 + x + 9
Derivative = 6x + 1
4x4 + 3x3 + x + 19
Derivative = 16x3 + 9x2 + 1
That’s all there is to taking the derivative of a polynomial!