AP Calculus AB Power rule — AP Calculus AB
1. Core Power Rule for Positive Integer Exponents ★★☆☆☆ ⏱ 4 min
The power rule is the most fundamental differentiation shortcut for AP Calculus AB, allowing you to compute derivatives of power functions without using the limit definition every time. A power function is any function of the form $f(x) = kx^n$, where $k$ is a constant coefficient and $n$ is any real number. This topic makes up 10-12% of Unit 2 exam score.
Exam tip: Always compute the general derivative first before plugging in the $x$-value for evaluation. Never substitute the $x$-value into the original function before differentiating, as this will always incorrectly give you a slope of zero.
2. Power Rule for Negative and Fractional Exponents ★★★☆☆ ⏱ 4 min
The power rule works for any real exponent, not just positive integers. This is one of the most frequently tested aspects of the rule on the AP exam, because it requires you to first rewrite radicals and reciprocals using exponent rules before applying differentiation.
Reciprocals $\frac{1}{x^k}$ rewrite to $x^{-k}$, and $k$-th roots $\sqrt[k]{x}$ rewrite to $x^{1/k}$. After rewriting, you apply the power rule exactly the same way as for positive integers.
Exam tip: Always explicitly write out your exponent subtraction step on paper for negative/fractional exponents. It is extremely common to accidentally subtract 1 from the coefficient instead of the exponent, or get the sign of the new exponent wrong.
3. Finding Tangent Lines with the Power Rule ★★★☆☆ ⏱ 3 min
A very common AP exam question asks you to find the equation of a tangent line to a power function curve at a given point. A tangent line to $y=f(x)$ at $x=a$ has two key properties: it passes through the point $(a, f(a))$, and its slope equals the derivative $f'(a)$, which you can compute quickly with the power rule. Once you have slope and a point, use point-slope form to write the final equation.
Exam tip: Never forget to calculate the $y$-coordinate from the original function. Many students only use the derivative to get slope and the given $x$-coordinate, leading to an incorrect $y$-intercept for the tangent line.
4. AP-Style Worked Practice Problems ★★★★☆ ⏱ 3 min
Common Pitfalls
Why: Forgets to multiply the original coefficient by the exponent, only subtracts 1 from the exponent.
Why: Incorrectly subtracts 1 from a negative exponent, resulting in a less negative exponent instead of a more negative one.
Why: Forgets that a constant term has a derivative of zero, since it can be written as $c x^0$.
Why: Forgets that the constant coefficient $\sqrt{2}$ needs to be multiplied into the derivative.
Why: Incorrectly multiplies the exponent with the coefficient, flipping the order of operations.