Differentiate Polynomials

Differentiating a polynomial function can help track the change of its slope. To differentiate a polynomial function, all you have to do is multiply the coefficients of each variable by their corresponding exponents, lower each exponent by one degree, and remove any constants. If you want to know how to break this down into a few easy steps, read on.

Steps

  1. Identify the variable terms and constant terms in the equation. A variable term is any term that includes a variable and a constant term is any term that has only a number without a variable. Find the variable and constant terms in this polynomial function: y = 5x3 + 9x2 + 7x + 3
    • The variable terms are 5x3, 9x2, and 7x
    • The constant term is 3
  2. Multiply the coefficients of each variable term by their respective exponents. Their products will form the new coefficients of the differentiated equation. Once you find their products, place the results in front of their respective variables. Here's how you do it:
    • 5x3 = 5 x 3 = 15
    • 9x2 = 9 x 2 = 18
    • 7x = 7 x 1 = 7
  3. Lower each exponent by one degree. To do this, simply subtract 1 from each exponent in each variable term. Here's how you do it:
    • 5x3 = 5x2
    • 9x2 = 9x1
    • 7x = 7
  4. Replace the old coefficients and old exponents with their new counterparts. To finish differentiating the polynomial equation, simply replace the old coefficients with their new coefficients and replace the old exponents with their values lowered by one degree. The derivative of constants is zero so you can omit 3, the constant term, from the final result.
    • 5x3 becomes 15x2
    • 9x2 becomes 18x
    • 7x becomes 7
    • The derivative of the polynomial y = 5x3 + 9x2 + 7x + 3 is y = 15x2 + 18x + 7
  5. Find the value of the new equation with a given "x" value. To find the value of "y" with a given "x," simply replace all of the "x"s in the equation with the given value of "x" and solve. For example, if you want to find the value of the equation at x = 2, simply plug the number 2 in place of every x in the equation. Here's how to do it:
    • 2 --> y = 15x2 + 18x+ 7 = 15 x 22 + 18 x 2 + 7 =
    • y = 60 + 36 + 7 = 103
    • The value of the equation at x = 2 is 103.



Tips

  • If you have negative or fractional exponents, don't worry! they follow the same rule. If for example you have x-1 it will become -x-2 and x1/3 becomes (1/3)x-2/3.
  • Remember that the definition of the derivative is: lim as h->0 of [f(x+h)-f(x)]/h
  • Finding indefinite integrals of polynomials is done the same way, only in reverse. Say you had 12x2 + 4x1 +5x0 + 0. Then you just add 1 to each exponent and divide by the new exponent. The result will be 4x3 + 2x2 + 5x1 + C, where C is a constant, since you can't tell what the value of the constant term will be.
  • Remember, this method only works when the exponent is a constant. For example, d/dx x^x is not x(x^(x-1))=x^x, but rather x^x(1+ln(x)). The power rule only applies to x^n for constant n.
  • This is known as the Power Rule of Calculus. It states:: d/dx[axn]=naxn-1

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