Key takeaway: Derivative rules are shortcut formulas, such as the constant, constant multiple, sum, power, product, quotient and chain rules, that find derivatives without using the limit definition each time.
Grade 12 – AP Calculus ABAP Calculus AB Unit 2Calculus
Definition of Derivative Rules
The derivative rules are a toolkit of formulas that turn differentiation into routine algebra. Each rule was proved once from the limit definition, so afterward a whole polynomial or product can be differentiated line by line.
The simplest rules handle constants, multiples and sums. Larger rules deal with powers, products, quotients and compositions. Picking the right rule depends on how the expression is built: added, multiplied, divided or nested.
Formula and notation
For constants \(c\) and differentiable functions \(f\) and \(g\):
\[ (c)^{\prime}=0,\quad (cf)^{\prime}=c\,f^{\prime},\quad (f\pm g)^{\prime}=f^{\prime}\pm g^{\prime},\quad (x^n)^{\prime}=n x^{n-1}. \]
The product, quotient and chain rules have their own pages.
Key facts
- The derivative of a sum is the sum of the derivatives, so polynomials go term by term.
- These derivative rules combine: one expression may need the sum, power and chain rules together.
- The product and quotient rules never reduce to multiplying or dividing the separate derivatives.
- Rewrite roots and reciprocals as powers, such as x to the 1/2, before applying the power rule.
Derivative Rules example
Differentiate \(g(x)=6x^5-4x^3+9x-11\) and evaluate \(g^{\prime}(1)\).
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Apply the sum rule, then the constant multiple and power rules to each term: \(6x^5\to 30x^4\) and \(-4x^3\to -12x^2\).
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The term \(9x\) becomes \(9\), and the constant \(-11\) becomes \(0\).
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So \(g^{\prime}(x)=30x^4-12x^2+9\), and \(g^{\prime}(1)=30-12+9=27\).
The derivative is \(g^{\prime}(x)=30x^4-12x^2+9\), and the slope at \(x=1\) is 27.
Common mistake
Watch out
A student differentiates \(h(x)=(x+4)(x-2)\) as \(1\cdot 1=1\), multiplying the two separate derivatives.
Why it is wrong
No derivative rule says the derivative of a product is the product of derivatives. Either expand first, \(h(x)=x^2+2x-8\), to get \(h^{\prime}(x)=2x+2\), or use the product rule: \(1\cdot(x-2)+(x+4)\cdot 1=2x+2\). Both give the same answer, and it is not 1. Learning the derivative rules means learning which one fits each shape.
Related terms and lessons
- AP Calculus AB · Definition of the Derivative and Basic RulesLessonWorksheetQuiz
- AP Calculus AB · Chain Rule, Implicit Differentiation and Inverse FunctionsLessonWorksheetQuiz
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