Key takeaway: The limit definition of the derivative defines the slope of a curve at a point as what the secant slope, the difference quotient, approaches when the step size h shrinks to zero.
Grade 12 – AP Calculus ABAP Calculus AB Unit 2Calculus
Definition of Limit Definition of the Derivative
The limit definition of the derivative builds the slope of a tangent line out of slopes of secant lines. Pick a point on a curve, pick a second point a horizontal distance \(h\) away, and find the slope between them. Then let \(h\) shrink toward zero.
For a smooth function the secant slopes settle toward one number. That number is the derivative. Every shortcut rule for differentiating is a consequence of this single limit.
Formula and notation
At any input \(x\), the derivative is
\[ f^{\prime}(x)=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}. \]
The fraction is the difference quotient. Many books write the horizontal step as \(h\) instead of \(\Delta x\). At one fixed input \(a\), replace \(x\) by \(a\) in the formula.
Key facts
- The difference quotient alone is a secant slope; only its limit as h goes to zero gives a tangent slope.
- Direct substitution of h = 0 gives 0 over 0, so the algebra must cancel h first.
- The limit definition of the derivative fails to give a number at a corner, where the two sides disagree.
- Shortcut rules such as the power rule are proved from this limit.
Limit Definition of the Derivative example
Apply the formula above to \(f(x)=3x^2-5x\) and find \(f^{\prime}(4)\).
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Evaluate \(f(4)=48-20=28\). Then \(f(4+h)=3(16+8h+h^2)-20-5h=28+19h+3h^2\).
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The difference quotient is \(\frac{19h+3h^2}{h}\).
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Cancel \(h\) to leave \(19+3h\), and let \(h\to 0\).
The expression \(19+3h\) approaches 19, so \(f^{\prime}(4)=\)19.
Common mistake
Watch out
A student starts the same problem by setting \(h=0\) immediately, gets \(\frac{28-28}{0}\), and writes that the derivative is 0.
Why it is wrong
Setting \(h=0\) too early produces the indeterminate form \(\frac{0}{0}\), which carries no information. Expand \(f(4+h)\), simplify the numerator and cancel the common factor \(h\) first. Only then substitute \(h=0\): the answer is 19, not 0. This order is the heart of the limit definition of the derivative.
Related terms and lessons
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