A derivative is not first a formula. It is the coefficient of the best first-order linear model of a function near one point.
School and Additional Mathematics rightly teach derivative rules because learners need efficient execution. BTT already has separate worked routes such as Calculus Rule Selection and Why Differentiation Feels Like a Collection of Rules. This guide owns the rigorous analysis layer underneath them.
Its central question is stronger than “what is f'(a)?” It asks what the existence of f'(a) guarantees about local behaviour, what continuity is forced, how global conclusions emerge from local derivative information, and how Taylor polynomials quantify approximation error.
Level: university-entry / undergraduate real-analysis enrichment. Prerequisite: the R25.01 guide on limits and continuity.
Reading route: difference quotients → derivative definition → local linearisation → differentiability implies continuity → derivative rules → inverse/implicit viewpoints → extrema → Rolle → Mean Value Theorem → monotonicity → convexity → Taylor polynomials → remainder → error control → counterexamples → practice → solutions.
1. The derivative is a limit of secant slopes
For a real-valued function f, the derivative at a is
f'(a)=lim_{h→0}[f(a+h)−f(a)]/h
when this finite limit exists.
The quotient is the slope of the secant line joining (a,f(a)) to (a+h,f(a+h)). The derivative is the limiting slope as the second point approaches the first.
2. Worked definition proof for x²
Let f(x)=x². Then
[f(a+h)−f(a)]/h=[(a+h)²−a²]/h=2a+h.
As h→0, this tends to 2a. Hence f'(a)=2a.
The derivative rule emerges from algebra plus a limit; the rule does not replace the definition.
3. Differentiability is equivalent to a first-order linear approximation
f is differentiable at a exactly when there exists a number A such that
f(a+h)=f(a)+Ah+r(h)
with r(h)/h→0 as h→0.
The coefficient A is f'(a).
This formulation explains “local linearity”. Near a, the dominant change is f'(a)h; the remainder is small compared with h itself.
4. Differentiability implies continuity
If f is differentiable at a, write
f(a+h)−f(a)=h·[f(a+h)−f(a)]/h.
As h→0, the first factor tends to 0 and the second tends to f'(a), so the product tends to 0.
Therefore f(a+h)→f(a): f is continuous at a.
The converse is false.
5. Continuity does not imply differentiability
For f(x)=|x| at zero, continuity is obvious.
But the right difference quotient is 1 while the left difference quotient is −1.
The derivative therefore does not exist at zero.
A corner is a geometric sign of failed first-order linear approximation, even when the function itself is perfectly continuous.
6. One-sided derivatives identify boundary behaviour
At endpoints or piecewise junctions, one can define right and left derivatives by restricting h to positive or negative values.
An ordinary derivative exists only when both one-sided derivatives exist and agree.
At a closed-interval endpoint, a one-sided derivative can still be useful even though the standard two-sided derivative is unavailable.
7. The product rule is a controlled decomposition of error
For differentiable f and g,
(fg)'(a)=f'(a)g(a)+f(a)g'(a).
The proof adds and subtracts f(a+h)g(a), then separates the difference quotient into two terms.
The continuity forced by differentiability ensures f(a+h)→f(a), allowing the limits to pass correctly.
8. The chain rule multiplies local linear models
If g is differentiable at a and f is differentiable at g(a), then
(f∘g)'(a)=f'(g(a))g'(a).
Conceptually, g converts a small input change h into approximately g'(a)h. Then f converts that intermediate change into approximately f'(g(a)) times it.
The product of the two scale factors is the derivative of the composition.
9. Derivatives of inverses are reciprocal slopes
If f is locally invertible near a, differentiable with f'(a)≠0, and the standard inverse-function hypotheses hold, then
(f⁻¹)'(f(a))=1/f'(a).
This is the one-dimensional shadow of the inverse function theorem.
The nonzero derivative condition matters: an inverse can exist while failing to have an ordinary finite derivative where f'(a)=0.
10. Fermat’s theorem gives a necessary condition for interior extrema
If f has a local maximum or minimum at an interior point a and f is differentiable at a, then f'(a)=0.
The converse is false. f(x)=x³ has f'(0)=0 but no local extremum at zero.
A zero derivative marks a stationary point, not automatically a maximum or minimum.
11. Rolle’s theorem converts equal endpoint values into a zero derivative
If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then some c∈(a,b) satisfies f'(c)=0.
Continuity on the closed interval ensures an extremum exists. Differentiability in the interior lets Fermat’s theorem apply.
Dropping either hypothesis can break the result.
12. The Mean Value Theorem turns an average slope into an instantaneous slope
If f is continuous on [a,b] and differentiable on (a,b), then there exists c∈(a,b) such that
f'(c)=[f(b)−f(a)]/(b−a).
The theorem says one tangent slope equals the secant slope across the full interval.
Its importance is larger than the picture: it turns local derivative bounds into global function bounds.
13. Derivative bounds imply Lipschitz bounds
If |f'(x)|≤M throughout an interval and the Mean Value Theorem applies between any two points x,y, then
|f(x)−f(y)|≤M|x−y|.
So a bounded derivative gives a global Lipschitz estimate.
This is a key analysis pattern: local rate control becomes global variation control.
14. Zero derivative on an interval forces constancy
If f'(x)=0 for every x in an interval, then f is constant on that interval.
For any a<b in the interval, the Mean Value Theorem gives f(b)−f(a)=f'(c)(b−a)=0.
The interval assumption matters: a function can be constant on separate components with different constants while having zero derivative wherever defined.
15. Sign of the derivative controls monotonicity
If f’≥0 on an interval, f is nondecreasing. If f’>0 everywhere, f is strictly increasing under the standard interval hypotheses.
If f’≤0, f is nonincreasing.
These conclusions again come from the Mean Value Theorem, not from reading a derivative graph informally.
16. A second derivative describes change of the first derivative
Where f” exists, it measures how f’ changes.
If f”≥0 on an interval, f’ is nondecreasing and f is convex there under standard smoothness hypotheses.
If f”≤0, the function is concave.
The second derivative test for stationary points is useful but only a sufficient criterion when f” has the required sign; f”(a)=0 is inconclusive.
17. Convexity can be defined without second derivatives
f is convex on an interval if for 0≤t≤1,
f((1−t)x+ty)≤(1−t)f(x)+tf(y).
The graph lies below its chords.
For differentiable convex functions, tangent lines lie below the graph. For twice differentiable functions, f”≥0 is a convenient sufficient-and-necessary condition on intervals.
18. Local linearisation gives first-order approximation
Near x=a,
f(x)≈f(a)+f'(a)(x−a).
For f(x)=√x near a=1, f(1)=1 and f'(1)=1/2, so
√(1+h)≈1+h/2.
At h=0.02 this gives 1.01, close to the true value because the neglected error is higher order than h.
19. Taylor polynomials extend linearisation to higher order
If enough derivatives exist, the degree-n Taylor polynomial about a is
T_n(x)=Σ_{k=0}^n f^{(k)}(a)(x−a)^k/k!.
It matches f and its first n derivatives at x=a.
Matching derivatives does not by itself guarantee that the infinite Taylor series equals f away from a. That requires additional convergence and remainder control.
20. Worked Taylor approximation for e^x
Because every derivative of e^x equals e^x, at a=0 all derivatives equal 1.
Thus
T_n(x)=1+x+x²/2!+⋯+x^n/n!.
The degree-3 approximation at x=0.1 is 1+0.1+0.005+0.000166…≈1.105166…
To know how reliable this is, we need a remainder theorem.
21. Lagrange’s remainder quantifies Taylor error
Under the standard differentiability hypotheses, Taylor’s theorem gives
f(x)=T_n(x)+f^{(n+1)}(ξ)(x−a)^{n+1}/(n+1)!
for some ξ between a and x.
If |f^{(n+1)}(t)|≤M on the relevant interval, then
|R_n(x)|≤M|x−a|^{n+1}/(n+1)!.
This converts an approximation into a certified error interval.
22. Worked error bound for e^0.1
Use the degree-3 Taylor polynomial about zero. On [0,0.1], e^t≤e^0.1<1.106.
Therefore
|R_3(0.1)|≤1.106·0.1⁴/4!≈0.00000461.
The approximation error is less than five millionths.
The theorem tells us why the approximation is good, not merely that a calculator agrees.
23. Taylor’s theorem is not the claim that every smooth function equals its Taylor series
A famous counterexample is
f(x)=e^{-1/x²} for x≠0, with f(0)=0.
This function is infinitely differentiable, and every derivative at zero equals zero.
Its Taylor series at zero is therefore identically zero, yet f(x)>0 for x≠0.
Smooth does not imply analytic.
24. Differentiability can fail even when slopes appear bounded
|x| has bounded one-sided slopes near zero but no derivative at zero.
x sin(1/x) is continuous at zero after defining f(0)=0, but its difference quotient sin(1/x) oscillates and the derivative at zero fails.
Local boundedness of a difference quotient is weaker than convergence of that quotient.
25. Derivatives can exist without being continuous
A function may be differentiable everywhere while f’ is discontinuous.
For example, define f(x)=x²sin(1/x) for x≠0 and f(0)=0.
Then f'(0)=0, while for x≠0, f'(x)=2x sin(1/x)−cos(1/x), which oscillates near zero.
So differentiability is not the same as continuously differentiable.
26. Darboux’s theorem gives derivatives an intermediate-value property
Even though derivatives need not be continuous, they cannot have jump discontinuities.
Darboux’s theorem states that derivatives have the intermediate value property.
If f’ takes values A and B at two points, it takes every value between A and B somewhere in between.
This is a subtle example of a property that resembles continuity without full continuity.
27. Newton’s method is local linearisation turned into an iteration
To solve f(x)=0, replace f near x_n by its tangent line:
f(x)≈f(x_n)+f'(x_n)(x−x_n).
Setting the linear model to zero gives
x_{n+1}=x_n−f(x_n)/f'(x_n).
Rapid convergence requires local hypotheses. Poor starting points, small derivatives, multiple roots or non-smooth behaviour can make the iteration slow, unstable or divergent.
28. Error propagation begins with the derivative
For a small perturbation Δx,
Δf≈f'(x)Δx.
This first-order model underlies sensitivity analysis, uncertainty propagation and condition-number ideas.
But approximation must be accompanied by a remainder estimate when the perturbation is not negligibly small or when curvature is large.
29. A dependable derivative-proof workflow
For existence, start from the difference quotient or the linear-remainder definition. Simplify before taking the limit.
For nonexistence, compare one-sided difference quotients or choose sequences of h-values producing incompatible limits.
For global conclusions, identify the interval and check the Mean Value Theorem hypotheses before using derivative signs or derivative bounds.
For Taylor approximation, specify centre a, degree n, derivative bounds and the interval containing the unknown ξ.
30. Common misconceptions
Continuous means differentiable. False: |x| at zero.
f'(a)=0 means a local extremum. False: x³ at zero.
The Mean Value Theorem only explains a tangent picture. Its main power is converting derivative information into global inequalities and monotonicity.
A Taylor polynomial is the function. It is an approximation unless a remainder argument shows equality in the limit.
Infinitely differentiable implies equal to the Taylor series. False.
31. Independent practice: twenty questions
- State the derivative definition at a.
- Use the definition to differentiate x².
- State the local-linearisation form of differentiability.
- Why does differentiability imply continuity?
- Why is |x| not differentiable at zero?
- State the product rule.
- State the chain rule.
- When does the reciprocal derivative formula for an inverse apply?
- State Fermat’s theorem for interior extrema.
- Give a stationary point that is not an extremum.
- State Rolle’s theorem.
- State the Mean Value Theorem.
- If |f’|≤7 on [a,b], what Lipschitz bound follows?
- If f’=0 on an interval, what follows?
- What does f’>0 imply on an interval?
- Write the degree-3 Taylor polynomial for e^x about zero.
- State the Lagrange remainder formula.
- Why does smooth not imply analytic?
- Can a derivative be discontinuous?
- Write Newton’s iteration for solving f(x)=0.
32. Worked solutions and checks
1. f'(a)=lim_{h→0}[f(a+h)−f(a)]/h when the finite limit exists.
2. [(a+h)²−a²]/h=2a+h→2a.
3. f(a+h)=f(a)+f'(a)h+r(h) with r(h)/h→0.
4. f(a+h)−f(a)=h times a difference quotient tending to f'(a); the product tends to zero.
5. The right difference quotient tends to 1 and the left to −1.
6. (fg)’=f’g+fg’.
7. (f∘g)'(a)=f'(g(a))g'(a).
8. Under local invertibility and differentiability hypotheses with f'(a)≠0; then (f⁻¹)'(f(a))=1/f'(a).
9. A differentiable function with a local interior maximum or minimum has derivative zero there.
10. f(x)=x³ at zero.
11. Continuous on [a,b], differentiable on (a,b), equal endpoint values imply some interior derivative is zero.
12. Continuous on [a,b] and differentiable on (a,b) imply an interior c with f'(c)=[f(b)−f(a)]/(b−a).
13. |f(x)−f(y)|≤7|x−y|.
14. f is constant on that interval.
15. f is strictly increasing under the standard interval hypotheses.
16. 1+x+x²/2+x³/6.
17. R_n(x)=f^{(n+1)}(ξ)(x−a)^{n+1}/(n+1)! for some ξ between a and x.
18. There are C∞ functions such as e^{-1/x²} extended by zero whose Taylor series at zero is zero but whose nearby values are not.
19. Yes; x²sin(1/x) extended at zero is differentiable while its derivative oscillates near zero.
20. x_{n+1}=x_n−f(x_n)/f'(x_n), where the denominator is nonzero and the local model is appropriate.
33. Where this guide hands off
R25.03 changes direction from local slope to global accumulation. Riemann sums, integrability and the Fundamental Theorem of Calculus will explain exactly when area-like accumulation and antiderivatives become two faces of one theory.
Sources and further study
Use MIT OpenCourseWare 18.100A Real Analysis for a rigorous analysis route, together with the preceding BTT Limits, Continuity and Sequences guide. School and A-Math derivative-rule routes remain separate application owners.