How Coefficients Affect Differentiation

By Mark Ryan

If the function you’re differentiating begins with a coefficient, the coefficient has no effect on the process of differentiation. You just ignore it and differentiate according to the appropriate rule. The coefficient stays where it is until the final step when you simplify your answer by multiplying by the coefficient.

Here’s an example: Differentiate y = 4x3.

Solution: You know by the power rule that the derivative of x3 is 3x2, so the derivative of 4(x3) is 4(3x2). The 4 just sits there doing nothing. Then, as a final step, you simplify: 4(3x2) equals 12x2. So

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(By the way, most people just bring the 3 to the front, like this:

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which gives you the same result.)

Here’s another example: Differentiate y = 5x.

Solution: This is a line of the form y = mx + b with m = 5, so the slope is 5, and thus the derivative is 5:

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(It’s important to think graphically like this from time to time.) But you can also solve the problem with the power rule:

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One final example: Differentiate

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Solution: The coefficient here is

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So, because

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(by the power rule),

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Keep in mind that pi, e, c, k, etc. are not variables! Don’t forget that

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are numbers, not variables, so they behave like ordinary numbers. Constants in problems, like c and k, also behave like ordinary numbers.

Thus, if

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This works exactly like differentiating y = 5x. And because

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is just a number, if

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This works exactly like differentiating y = 10. You’ll also see problems containing constants like c and k. Be sure to treat them like regular numbers. For example, the derivative of y = 5x + 2k3 (where k is a constant) is 5, not 5 + 6k2.