Derivative Rules
In short: Derivative rules are pattern-matching shortcuts that give the derivative of constants, powers, constant multiples, sums and differences without going back to a limit each time.
AP Calculus ABCalculusAP Calculus AB Unit 2
What is Derivative Rules?
Running the limit process for every function would eat the whole exam, so a handful of derivative rules were proved once; now you just recognize the pattern in front of you and apply the matching one.
The basic four are: a constant has derivative 0; \(\frac{d}{dx}[x^n] = nx^{n-1}\); a constant factor rides along, \(\frac{d}{dx}[c\,f(x)] = c\,f^{\prime}(x)\); and a sum or difference is differentiated term by term. Products, quotients and nested functions need their own, separate rules.
How to spot it
- The function is built from terms added or subtracted, each one a number times a power of x.
- A square root or a fraction with x in the denominator is hiding a power that can be rewritten.
- The problem asks for dy/dx of a polynomial, or for a slope at one input of such a function.
- No two expressions containing x are multiplied or nested, so the basic derivative rules are enough.
The method
- Rewrite roots and reciprocals as powers: \(\sqrt{x} = x^{1/2}\) and \(\frac{1}{x^3} = x^{-3}\).
- Split the expression at each plus or minus sign and handle every term on its own.
- For a term \(c\,x^n\), multiply \(c\) by \(n\) and lower the exponent by 1. A bare number goes to 0.
- Reassemble the pieces, as the derivative rules for sums allow, and if you like rewrite negative or fractional exponents as fractions and roots.
Derivative Rules: solved example
Problem
Find \(f^{\prime}(x)\) for \(f(x) = 6x^4 – 5\sqrt{x} + \frac{9}{x^2} – 8\).
Solution
Rewrite with exponents: \(f(x) = 6x^4 – 5x^{1/2} + 9x^{-2} – 8\).
Term by term: \(24x^3\); then \(-5 \cdot \frac{1}{2}x^{-1/2} = -\frac{5}{2}x^{-1/2}\); then \(9 \cdot (-2)x^{-3} = -18x^{-3}\); the constant \(-8\) gives 0.
Result: \(f^{\prime}(x) = 24x^3 – \frac{5}{2\sqrt{x}} – \frac{18}{x^3}\).
Try it
Question 1
Differentiate \(y = 7x^3 – 2x + 15\).
Show the answer
\(\frac{dy}{dx} = 21x^2 – 2\), because the constant 15 disappears.
Question 2
Find the slope of \(y = \frac{4}{x}\) at \(x = 2\).
Show the answer
Write \(y = 4x^{-1}\), so \(y^{\prime} = -4x^{-2} = -\frac{4}{x^2}\). At \(x = 2\) the slope is \(-\frac{4}{4} = -1\).
Derivative Rules vs. Product Rule
Sums split cleanly: the derivative of \(f + g\) is \(f^{\prime} + g^{\prime}\). Products do not behave that way. The derivative of \(x^2 \cdot x^3\) is not \(2x \cdot 3x^2\); you need the product rule, or you can first simplify to \(x^5\).
Keep going
Lessons
Worksheets and quizzes
- Power, Sum and Basic Function Derivative Rules – Worksheet
- Power, Sum and Basic Function Derivative Rules – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus