Laboratory · Derivatives
Derivatives: from secant to tangent
Fix a point, bring nearby points closer and compare slopes: when does a tangent emerge, and when does the derivative fail to exist?
Choose an example or enter a function.
A derivative formula value or a finite difference does not alone certify differentiability. The formula may need simplifying; failure to evaluate may reflect domain or precision. A line drawn in free mode is a candidate requiring justification.
Height: the graph of f
fLeft secantRight secantTangent / candidate
Slope: the graph of f′
f′f″
From the left
From the right
Value f(a)
Slope f′(a)
What happens at a
Derivative formulas
Tangent line
Case explanation
At regular points of this example, both quotients approach the derivative value. The analytic formulas apply on the stated domain.
A symmetric quotient alone does not prove differentiability: for |x| at zero it is 0, but the one-sided quotients differ.
Compare quotients as |h| decreases
Approximate numerical values. h is never set to zero.