Calculus: Derivatives · Grades 12–12

Derivatives (Calculus) Worksheets

Grade 12 worksheets differentiating polynomials with the power rule, plus sinusoidal, exponential, rational, and radical functions, each its own short rule built from the power rule.

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What it is

The derivative measures how a function changes. For polynomials, the power rule says the derivative of a·xⁿ is a·n·xⁿ⁻¹. Rational (a/xⁿ) and radical (a√x) functions use the exact same power rule once they're rewritten as a·x to a power. Sinusoidal and exponential functions each have their own short rule, with an extra factor of k whenever the angle or exponent is k·x instead of plain x.

Why it matters

Derivatives are the heart of calculus, rates of change, slopes of curves, and optimisation across science, engineering, and economics.

How to do it

  1. Differentiate each term separately.
  2. Polynomial, rational, or radical: multiply the coefficient by the exponent, then reduce the exponent by 1 (a/xⁿ is x to the power −n; a√x is x to the power 1/2).
  3. Sinusoidal: d/dx[sin(kx)] = k·cos(kx); d/dx[cos(kx)] = −k·sin(kx).
  4. Exponential: d/dx[e^(kx)] = k·e^(kx).
  5. The derivative of a constant term is 0.

Examples by level

  • Beginnerd/dx (3x³ + 5) =9x²
  • Intermediated/dx (6sin⁡(2x)6\sin(2x)) =12cos⁡(2x)12\cos(2x)
  • Advancedd/dx (−2x⁴ − 9x³) =−8x³ − 27x²

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

  • Forgetting the constant term differentiates to 0.
  • Not reducing the exponent by 1.
  • Dropping the coefficient when multiplying by the power.
  • Differentiating cos(kx) and forgetting the negative sign that comes with it.
  • Forgetting the chain-rule factor of k when the angle or exponent is k·x rather than plain x.

Tips

  • Do one term at a time.
  • Constants vanish; a plain x becomes 1.
  • Watch the sign: cos differentiates to −sin, but sin differentiates to +cos.

For parents

The power rule is mechanical: multiply by the exponent, then subtract 1 from it, one term at a time. The sinusoidal, exponential, rational, and radical rules are each a short variation on that same idea.

For teachers

Have students verbalise 'bring down the power, reduce the power' for each term until it's automatic, then layer in the four new function families one at a time rather than mixed together.

Key vocabulary

derivative
the instantaneous rate of change
power rule
d/dx xⁿ = n·xⁿ⁻¹

Frequently asked questions

What is the power rule?
The derivative of a·xⁿ is a·n·xⁿ⁻¹, multiply by the exponent and lower the exponent by one.
What is the derivative of a constant?
Zero, a constant doesn't change, so its rate of change is 0.
What is the derivative of sin(kx) and cos(kx)?
d/dx[sin(kx)] = k·cos(kx), and d/dx[cos(kx)] = −k·sin(kx). The extra factor of k comes from the chain rule.
What is the derivative of e^(kx)?
d/dx[e^(kx)] = k·e^(kx), the exponential function is its own derivative, up to that chain-rule factor of k.

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