Higher Derivatives, Rolle's Theorem and MVT | NIMCET

Quick answer: The second derivative $\frac{d^2y}{dx^2}$ is the derivative of $\frac{dy}{dx}$. Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b) and $f(a) = f(b)$, then $f'(c) = 0$ for some c in (a, b). Lagrange's mean value theorem (MVT): if f is continuous on [a, b] and differentiable on (a, b), then $f'(c) = \frac{f(b) - f(a)}{b - a}$ for some c in (a, b).

Rolle's theorem and the MVT are listed in the revised NIMCET calculus syllabus (see the NIMCET syllabus update).

Higher-order derivatives

Parametric second derivative:

$\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx}$

It is not $\frac{d^2y/dt^2}{d^2x/dt^2}$.

nth derivatives of standard functions

Rolle's theorem

Geometric meaning: somewhere between two points at the same height, the tangent is horizontal.

Consequence: between two roots of $f(x) = 0$ there is at least one root of $f'(x) = 0$.

Lagrange's mean value theorem

Geometric meaning: some tangent is parallel to the chord joining the endpoints.

Rolle's theorem is the special case $f(a) = f(b)$.

For a quadratic $f(x) = px^2 + qx + r$, the MVT point is always the midpoint: $c = \frac{a + b}{2}$.

Worked examples

Example 1: second derivative

If $y = x^3 - 3x^2 + 5$, find $y''$.

$y' = 3x^2 - 6x$ and $y'' = 6x - 6$.

Example 2: implicit second derivative

If $x^2 + y^2 = 1$, find $y''$.

Step 1: $y' = -\frac{x}{y}$.

Step 2: $y'' = -\frac{y - xy'}{y^2} = -\frac{y + \frac{x^2}{y}}{y^2} = -\frac{x^2 + y^2}{y^3} = -\frac{1}{y^3}$.

Example 3: Rolle's theorem

Verify Rolle's theorem for $f(x) = x^2 - 4x + 3$ on [1, 3] and find c.

Step 1: f is a polynomial, and $f(1) = f(3) = 0$.

Step 2: $f'(c) = 2c - 4 = 0$ gives $c = 2 \in (1, 3)$ ✓.

Example 4: MVT

Find c for $f(x) = x^3$ on [0, 3].

Step 1: $\frac{f(3) - f(0)}{3} = 9$.

Step 2: $3c^2 = 9$, so $c = \sqrt{3} \in (0, 3)$.

Example 5: when the theorem fails

Does Rolle's theorem apply to $f(x) = \lvert x\rvert$ on [−1, 1]?

No. Although $f(-1) = f(1)$, f is not differentiable at 0, and indeed $f'(x) = \pm 1$ is never 0.

Summary table

Exam tips

• Before applying either theorem, check continuity and differentiability; questions often hinge on a failed condition.
• For quadratics, write c as the midpoint immediately.
• Rolle's theorem helps count roots: between two roots of f lies a root of f′.

Next, read applications of derivatives: tangents and normals, or see the NIMCET Calculus chapter.

Common mistakes

• Using $\frac{d^2y/dt^2}{d^2x/dt^2}$ for the parametric second derivative.
• Applying Rolle's theorem without checking $f(a) = f(b)$.
• Accepting a value of c that lies outside (a, b).
• Ignoring points where f is not differentiable.

Frequently Asked Questions

What is Rolle's theorem?

If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f′(c) = 0 for some c in (a, b).

What is Lagrange's mean value theorem?

If f is continuous on [a, b] and differentiable on (a, b), then f′(c) = (f(b) − f(a))/(b − a) for some c in (a, b).

What is the geometric meaning of the mean value theorem?

There is a point where the tangent is parallel to the chord joining the endpoints.

What is the nth derivative of sin x?

sin(x + nπ/2).

Does Rolle's theorem apply to |x| on [−1, 1]?

No, because |x| is not differentiable at 0.