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CalculusBy Educator Today

C07 · Unit C2 Derivative foundations

Power rules, constants and elementary derivatives

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 2.5–2.7
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

For differentiable terms, derivatives distribute over sums and constant multiples. The power rule gives d(x to the power n)/dx equals nx to the power nminus 1 where the function and rule are defined. Basic derivatives include sin x tends to cos x, cos x tends to minus sin x, eˣ tends to eˣ and ln x tends to 1 over x for x>0. Trigonometric derivative formulas assume radians.

Worked example

If f(x) equals 4x cubed minus 2x+sin x, then f minutes (x) equals 12x squared minus 2+cos x. At zero f minutes (0) equals minus 1. The constant derivative and the cosine value must both be included.

Explore the mathematics

Power rules, constants and elementary derivatives

Bring two points together and compare secant and tangent slopes.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict how the secant changes as the two points get closer.xy-1123At x=1, h=0.125; secant slope=3.3906

Predict before continuing

Predict how the secant changes as the two points get closer.

Read the complete explanation
  1. Predict how the secant changes as the two points get closer.
  2. The orange secant joins two points: its slope is Δy/Δx.
  3. The green tangent has slope 3. Reduce h to see the secant approach it.
  4. The derivative is a limit, not substitution of h=0 into a quotient with zero denominator. Here f′(1)=3.
See the idea: Power rules, constants and elementary derivatives
See the idea: Power rules, constants and elementary derivativesThe curve is f(x)=4x³−2x+sin x. At zero its derivative is −1, so y=−x is its tangent. The sine contribution adds cos(0)=1 to the polynomial derivative −2.xy-1-6-0.5-3000.5316

The curve is f(x)=4x³−2x+sin x. At zero its derivative is −1, so y=−x is its tangent. The sine contribution adds cos(0)=1 to the polynomial derivative −2.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

The derivative of a constant is zero. The power rule reduces the exponent by one; it does not merely multiply the original term.

Investigate and explain

Differentiate a polynomial two ways: term by term and through the definition. Explain why a rule compresses, rather than replaces, the limiting idea.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.