Master the fundamental shortcuts for differentiation by seeing how each rule
emerges naturally from the limit definition.
While we can always find derivatives using the limit definition, doing so for every
function would be tedious. Fortunately, we can derive general rules that act as
shortcuts. In this lesson, you'll see where these rules come from and how to remember them.
1. The Limit Definition (Review)
Every derivative rule starts from the same foundation: the limit definition of the derivative.
Definition: The derivative of f(x) is:
This represents the instantaneous rate of change—the slope of the tangent line at a point.
Let's see how this definition gives us our first rule.
2. The Constant Rule
Let's find the derivative of a constant function: f(x) = c, where c is any constant number.
The Constant Rule:
💡 Remember: "Flat lines don't change" — A horizontal line has zero slope everywhere.
3. The Power Rule
Now for the most important rule: the derivative of xn. Let's derive it from
the limit definition using the binomial theorem.
The Power Rule:
💡 Remember: "Pull down the power, subtract one" — The exponent comes down as a coefficient,
then decrease the exponent by 1.
Examples:
4. The Constant Multiple Rule
What happens when we multiply a function by a constant? Let's find the derivative of k·f(x).
The Constant Multiple Rule:
💡 Remember: "Constants are passengers" — They come along for the ride but don't affect
the differentiation process.
Example:
5. The Sum and Difference Rules
Our final basic rule: the derivative of a sum (or difference) is the sum (or difference)
of the derivatives.
The Sum/Difference Rules:
💡 Remember: "Differentiate term by term" — Take the derivative of each piece separately,
keeping the addition or subtraction.
🎯 Putting It All Together: These four rules are the foundation of all basic
differentiation. You can now find the derivative of any polynomial!