A reliable method
Rewrite radicals and reciprocals as powers when their domains permit it. Apply the power rule and evaluate only after differentiating.
Worked foundation example
If f(x)=2x³+4, find f′(3). The power rule gives f′(x)=6x². The constant term differentiates to zero. Substitute 3: f′(3)=54.
The common trap
The derivative of x raised to a power changes both the coefficient and exponent. A negative power does not automatically give a positive derivative.
Connect the representations
A derivative value gives tangent slope, not the height of the function. Keep these two quantities distinct in a tangent-line problem.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare output with instantaneous change
For f(x)=3x²+4, f(2)=16 whereas f′(2)=12. The input two is shared, but the operations are different. Differentiating the function before substituting preserves the nearby behavior needed for a derivative. Substituting first produces the constant 16, whose derivative is zero, but that is a different function. A derivative question must preserve the variable until the differentiation is complete.
Transfer and explain
Transfer: f(x)=5√x at x=4 has output 10 and derivative 5/4. Rewrite the radical as a power, show the factor 1/2, and explain why the two quantities need not agree.
If f(x)=2x³+4, find f′(3).
- The power rule gives f′(x)=6x².
- The constant term differentiates to zero.
- Substitute 3: f′(3)=54.
Compare the other answer choices
Lower the exponent but omit its factor.
Drop the leading coefficient of the original function.
Evaluate the original function instead of its derivative.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus