From second derivatives to nth order, from motion analysis to concavity and beyond — this worksheet provides a rigorous, concept‑deep exploration of higher order derivatives. Designed for AP Calculus AB/BC, university Calculus I & II, and independent learners who want more than just computation.
Introduction
When Newton and Leibniz first laid the foundations of calculus, they understood that change itself could change. Velocity (the first derivative of position) could increase or decrease, and that rate of change – acceleration – is a derivative of a derivative. This simple yet profound idea opens the door to higher order derivatives: the second derivative \(f''(x)\), third derivative \(f'''(x)\), and beyond.
Why stop at the first derivative? In physics, acceleration is only the beginning; the third derivative, jerk, describes how acceleration changes, critical in designing smooth rides for elevators and roller coasters. In economics, the second derivative of a cost function tells us whether marginal cost is increasing or decreasing – essential for optimisation. In beam theory, the fourth derivative of deflection relates to the load distribution. Higher order derivatives are everywhere once you start looking.
This worksheet is for students who have mastered basic differentiation (power, product, quotient, chain rule) and are ready to deepen their understanding. You’ll learn to compute second, third and nth derivatives, interpret their meaning, and apply them to real‑world models. You’ll also encounter conceptual questions that test your grasp of what these derivatives represent.
Mastery here means you can: compute any order derivative efficiently, recognise patterns (polynomials vanish, exponentials repeat, trig functions cycle), apply the Leibniz rule for products, and – most importantly – explain what the second derivative tells you about a function’s graph and about a physical process. Let’s begin.
Learning Objectives
Compute second, third and higher order derivatives using all differentiation rules.
Interpret \(f''(x)\) in terms of concavity, acceleration, and rate of change of slope.
Find inflection points and intervals of concavity using the second derivative.
Apply higher order derivatives to motion (position, velocity, acceleration, jerk).
Combine product, quotient, and chain rules when computing \(f''(x)\).
Use Leibniz notation \(\frac{d^ny}{dx^n}\) and understand its meaning.
Derive formulas for nth derivatives of simple functions (\(x^n\), \(e^{ax}\), \(\sin(ax)\), \(\cos(ax)\)).
Recognise and correct common errors in higher order differentiation.
Conceptual Foundation
2.1 Definition and Notation
If \(y = f(x)\) is differentiable, its derivative \(f'(x)\) (or \(\frac{dy}{dx}\)) is a new function. If \(f'\) is itself differentiable, we define the second derivative as:
Similarly, the third derivative \(f'''(x) = \frac{d^3y}{dx^3}\), and the \(n\)-th derivative is denoted \(f^{(n)}(x)\) or \(\frac{d^ny}{dx^n}\).
Interpretation: \(f''(x)\) measures the rate of change of the slope. Geometrically, it tells us whether the graph is concave up (\(f''>0\)) or concave down (\(f''<0\)). In motion, if \(s(t)\) is position, \(s''(t)\) is acceleration.
2.2 When to Use Higher Order Derivatives
Concavity & inflection points – find where \(f''(x)=0\) and changes sign.
Motion analysis – from position to velocity to acceleration to jerk.
Optimisation – second derivative test for local maxima/minima.
Taylor series – requires knowledge of all derivatives at a point.
Differential equations – many physical laws involve second derivatives (Newton’s \(F = m a\)).
When NOT to use them: If you only need slope, stop at first derivative. For linear functions, higher derivatives are zero – but that’s still information.
2.3 Structural Recognition & Efficiency
Before differentiating twice, look for patterns:
Polynomials: each derivative lowers the degree; eventually reach zero.
Products: use the Leibniz rule for nth derivative: \((fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(k)} g^{(n-k)}\).
Professor’s advice: When computing \(f''(x)\), always simplify \(f'(x)\) first. Many students carry messy algebra into the second derivative and make sign errors. Factor, cancel, rewrite negative exponents – it pays off.
Interpret your answers in context. Include units where appropriate.
The position of a particle (in metres) is \(s(t) = t^3 - 6t^2 + 9t + 2\) for \(t \ge 0\). Find velocity, acceleration, and jerk at \(t=2\). When is the particle decelerating?
A capacitor charge follows \(q(t) = 5e^{-2t}\cos(3t)\) coulombs. Find the current \(i(t) = q'(t)\) and the rate of change of current \(i'(t)\).
The deflection of a beam is \(y(x) = 4\sin\!\left(\frac{\pi x}{3}\right)\) mm. Find the second derivative \(y''(x)\) (proportional to bending moment). Evaluate at \(x=1\).
For a company, the cost function is \(C(x) = 0.001x^3 - 0.3x^2 + 50x + 1000\) dollars for \(x\) units. Find marginal cost \(C'(x)\) and the rate of change of marginal cost \(C''(x)\). Interpret \(C''(100)\).
A population grows as \(P(t) = \frac{1000}{1+9e^{-0.2t}}\). Find \(P''(t)\) and determine when the growth rate is fastest (inflection point).
In an RL circuit, the current is \(I(t) = \frac{V}{R}(1 - e^{-Rt/L})\). Find \(dI/dt\) and \(d^2I/dt^2\). Interpret the sign of the second derivative.
For an object in simple harmonic motion, \(x(t) = A\cos(\omega t + \phi)\). Show that \(x''(t) = -\omega^2 x(t)\).
The gas law \(PV = nRT\) with constant \(n,R\). If \(P\) and \(V\) are functions of \(t\), find \(\frac{d^2P}{dt^2}\) in terms of derivatives of \(V\).
A ladder 10 m long leans against a wall; its foot slides away at 1 m/s. Find the second derivative of the height of the top with respect to time at the instant the foot is 6 m from the wall.
The temperature in a metal rod is \(T(x) = 80e^{-0.1x}\) °C. Find \(T''(x)\) and interpret its meaning.
Part E — Exam‑Level & Mixed Detection (61–75)
Multiple choice, error diagnosis, and rule‑identification. Circle or explain.
If \(f(x) = (x^2+1)^4\), then \(f''(0) =\) (A) 0 (B) 4 (C) 8 (D) 12
The second derivative of \(y = \ln(\cos x)\) is: (A) \(-\sec^2 x\) (B) \(\sec^2 x\) (C) \(-\csc^2 x\) (D) \(\csc^2 x\)
For which function does \(f''(x) = f(x)\)? (A) \(e^{2x}\) (B) \(\sin x\) (C) \(\cosh x\) (D) \(\ln x\)
Given \(f(1)=2\), \(f'(1)=-3\), \(f''(1)=4\), find \(g''(1)\) if \(g(x) = x^2 f(x)\).
A student writes: "For \(f(x) = x^2 e^x\), \(f''(x) = 2e^x + 2x e^x + x^2 e^x = e^x(x^2+2x+2)\)." Is this correct? If not, identify the mistake.
Find the error: For \(f(x) = \sin(x^2)\), \(f'(x) = \cos(x^2)\), \(f''(x) = -\sin(x^2)\).
If \(y = \tan x\), which statement is true? (A) \(y'' = 2\sec^2 x \tan x\) (B) \(y'' = 2\sec^2 x\) (C) \(y'' = 2\sec x \tan x\) (D) \(y'' = \sec^2 x \tan x\)
Determine whether the following reasoning is valid: "If \(f''(c) = 0\), then \(c\) is an inflection point." Provide a counterexample.
Which rules are needed to differentiate \(f(x) = e^{x^2} \sin(3x)\) twice? List them.
Given \(f(x) = \frac{x}{x^2+1}\), find \(f''(1)\).
If \(h(t) = \sqrt{t+1}\), find \(h''(3)\).
Let \(F(x) = f(g(x))\) with \(f(u) = u^3\), \(g(x) = \cos x\). Find \(F''(x)\).
Suppose \(f''(x) = 6x\) and \(f'(0)=2\), \(f(0)=1\). Find \(f(x)\).
Part F — Challenge & Extension (76–85)
For advanced students: nth derivatives, Leibniz rule, proofs.
Find a general formula for \(f^{(n)}(x)\) if \(f(x) = \frac{1}{ax+b}\).
Use induction to prove that \(\frac{d^n}{dx^n} e^{ax} = a^n e^{ax}\).
For \(f(x) = \sin^2 x\), find \(f^{(n)}(x)\) in terms of sine/cosine.
Use Leibniz rule to find the 10th derivative of \(x^3 \cos x\).
Find \(f^{(4)}(x)\) for \(f(x) = \ln(1+x^2)\).
If \(y = \arctan x\), show that \((1+x^2)y'' + 2x y' = 0\).
Given \(f(x) = x^{n-1} \ln x\), find \(f^{(n)}(x)\).
Let \(f\) be twice differentiable and \(g(x) = f(\sqrt{x})\). Find \(g''(x)\) in terms of \(f'\) and \(f''\).
Find the error in the "proof" that all derivatives of \(x^2\) are zero for \(x>0\): "\(x^2 = x\cdot x\), derivative \(=1\cdot x + x\cdot 1 = 2x\); second derivative = 2; third derivative = 0; all further derivatives = 0." Actually \(x^2\) is a polynomial; is there an error?
Prove that if \(f\) is even and twice differentiable, then \(f'\) is odd and \(f''\) is even.
55. \(P''(t)=\frac{1800e^{-0.2t}(9e^{-0.2t}-1)}{(1+9e^{-0.2t})^3}\); max growth rate when \(9e^{-0.2t}=1 \Rightarrow t=5\ln 3\)
56. \(I'(t)=\frac{V}{L}e^{-Rt/L}\); \(I''(t)= -\frac{VR}{L^2}e^{-Rt/L}\) (negative, so current increase slows)
57. Proof in solution
58. \(\frac{d^2P}{dt^2}= \frac{2(V')^2 - V V''}{V^2} nRT\) – see solution
59. \(\frac{d^2y}{dt^2} = -\frac{125}{64}\) m/s² at that instant.
60. \(T''(x)=0.8e^{-0.1x}\) (positive, so temperature decreasing at an increasing rate? actually \(T'\) negative, \(T''\) positive means rate of decrease slows)
A higher order derivative is the derivative of a derivative. The second derivative measures the rate of change of the slope; the third derivative measures the rate of change of acceleration (jerk), and so on.
How do I find the second derivative of a product?
Differentiate once (using product rule), then differentiate the result – again using product/chain rules as needed. You can also use the formula \((fg)'' = f''g + 2f'g' + fg''\).
What is an inflection point and how is it related to the second derivative?
An inflection point is where the concavity changes, i.e. where \(f''(x)\) changes sign. Typically \(f''(x)=0\) at that point, but zero alone is not sufficient; the sign must change.
Are higher order derivatives used on the AP Calculus exam?
Yes, especially second derivatives for concavity, inflection points, and motion problems. Third derivatives appear occasionally (jerk).
What is Leibniz notation for higher derivatives?
\(\frac{d^2y}{dx^2}\), \(\frac{d^3y}{dx^3}\), etc. It emphasises the operator aspect: \(\frac{d}{dx}\) applied twice.
How can I check if my second derivative is correct?
Use a graphing calculator or online tool to compare numerical values at a few points, or differentiate using alternative methods (e.g. simplify first).
What’s the most common mistake when computing \(f''\)?
Forgetting to apply the chain rule again when differentiating a composite function that already appeared in \(f'\). Also algebraic simplification errors.
Can I find a formula for the nth derivative of a function?
For many elementary functions yes – exponentials, sines/cosines, polynomials, rational functions (using patterns or Leibniz rule).
Exam Psychology & Strategic Advice
AP / University exam traps:
❌ Forgetting the chain rule in the second derivative – especially with trigonometric or exponential inner functions.
❌ Misinterpreting \(f''(x)=0\) as automatically an inflection point (check sign change!).
❌ Sign errors when differentiating negative exponents or using quotient rule twice.
❌ Not simplifying \(f'(x)\) before differentiating again – leads to messy algebra.
How graders award points: They look for correct first derivative (often partial credit), correct application of rules in second derivative, and final simplified form. Show each step clearly.
🧮
✨ Free Tool — 2026
Want More Worksheets? Generate Unlimited Free Math Worksheets!
Instantly create printable worksheets on any topic — derivatives, algebra, fractions & more. Includes full answer keys. No signup needed.