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What are the rules for differentiating polynomials?

Published in Calculus Differentiation 3 mins read

The primary rule for differentiating polynomials is the power rule, which states that the derivative of xn is nxn-1. Combining this with the constant multiple rule and the sum/difference rule, we can differentiate any polynomial.

Rules for Differentiating Polynomials Explained:

Here's a breakdown of the rules, along with examples:

1. The Power Rule

  • Statement: If f(x) = xn, then f'(x) = nxn-1
  • Explanation: Multiply the original exponent (n) by the term and then reduce the exponent by 1.
  • Example:
    • If f(x) = x3, then f'(x) = 3x2
    • If f(x) = x7, then f'(x) = 7x6
    • If f(x) = x1 (or simply x), then f'(x) = 1x0 = 1.

2. The Constant Multiple Rule

  • Statement: If g(x) = k f(x), where k is a constant, then g'(x) = k f'(x).
  • Explanation: The derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of the function.
  • Example:
    • If g(x) = 5x2, then g'(x) = 5 * (2x) = 10x
    • If g(x) = -3x4, then g'(x) = -3 * (4x3) = -12x3

3. The Constant Rule

  • Statement: If f(x) = k, where k is a constant, then f'(x) = 0.
  • Explanation: The derivative of any constant is zero. This makes sense because a constant function has a slope of zero.
  • Example:
    • If f(x) = 7, then f'(x) = 0
    • If f(x) = -2, then f'(x) = 0

4. The Sum and Difference Rule

  • Statement: If h(x) = f(x) + g(x) or h(x) = f(x) - g(x), then h'(x) = f'(x) + g'(x) or h'(x) = f'(x) - g'(x), respectively.
  • Explanation: To differentiate a sum or difference of terms, you can differentiate each term individually and then add or subtract the results.
  • Example:
    • If h(x) = x3 + 2x2 - x + 5, then h'(x) = 3x2 + 4x - 1 + 0 = 3x2 + 4x - 1

Summary Table:

Rule Function Derivative Example
Power Rule xn nxn-1 f(x) = x4, f'(x) = 4x3
Constant Multiple Rule k * f(x) k * f'(x) g(x) = 2x3, g'(x) = 6x2
Constant Rule k 0 f(x) = 9, f'(x) = 0
Sum/Difference Rule f(x) ± g(x) f'(x) ± g'(x) h(x) = x2 + x, h'(x) = 2x + 1

By applying these rules systematically, you can find the derivative of any polynomial function.

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