How to Apply the Constant Multiple Rule of a Derivative
- 1). Know that the power rule for positive integers states that if "n" is a positive integer then d/dx (x^n) = nx^(x-1).
- 2). Remember this rule in words: Subtract 1 from the original exponent, and multiply the result by the original exponent "n".
- 3). Memorize the constant multiple rule: d/dx (cu) = (c) du/dx. In words, if "u" is differentiable and "c" is a constant, then the derivative of (cu) is the unchanged constant "c" times the derivative of "u".
- 4). Prove this rule: d/dx (cu) equals the limit of h as it goes to 0 for [cu (x+h) - cu (x)]/h equals (c) times limit as h goes to 0 for [u (x+h) - u (x)]/h equals (c) du/dx.
- 5). Look at a couple of examples to anchor the concept: d/dx (4x^2) = 4 times 2x = 8x.
- 6). Note a special case for the constant multiple rule: the derivative of a negative differentiable "u" is the same as the negative of its derivative. Here's an example: d/dx (-u) = d/dx (-1 times u) = (-1) times d/dx (u) = (- du/dx).
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