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Keeping a constant or forgetting the power-rule coefficient

A derivative measures change. Bring down each exponent and reduce it by one; a constant contributes zero.

Spot the wrong turn

Differentiate f(x) = 2x³ + 4x − 9.

Incorrect answer: f′(x) = 2x² + 4x − 9

A constant disappears because it does not change with x; it does not become one. Also, differentiating a product of two functions generally needs the product rule, not a product of derivatives.

Work it through

The correction

f′(x) = 6x² + 4

Apply the power rule to each term and drop the constant.

  1. Multiply each coefficient by its exponent.
  2. Reduce that exponent by one; the derivative of a constant is zero.
  3. Combine the derivatives of the terms, then substitute a point only if a numerical slope is requested.

Try the reasoning in another example

Differentiate g(x) = 3x² − 5x + 7.

  1. The derivative of 3x² is 6x; the derivative of −5x is −5.
  2. The constant 7 contributes zero.

Answer: g′(x) = 6x − 5

Explain why each step preserves the quantity or relationship. If you can compute the answer but cannot explain the step, revisit the model before adding more questions.

The short check below reuses the first example as a retrieval question and adds a second question. It checks this method; it does not establish a grade level.

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1 Differentiate f(x) = 2x³ + 4x − 9.
2 Find the slope of y = x² at x = 5.