Unit 3.6 – Calculating Higher-Order Derivatives

AP® Calculus AB & BC | Second, Third, and nth Derivatives

Higher-order derivatives help us analyze how a function's behavior changes over time. The first derivative is about slope or velocity, the second is about curvature (concavity, acceleration), and higher ones can describe jerk or snap in physics and more! For all, just keep differentiating one more time.

🔢 Notation for Higher-Order Derivatives

Order	Lagrange Notation	Leibniz Notation	Meaning
First	$$f'(x)$$	$$\frac{dy}{dx}$$	Slope/Velocity
Second	$$f''(x)$$	$$\frac{d^2y}{dx^2}$$	Concavity/Acceleration
Third	$$f'''(x)$$	$$\frac{d^3y}{dx^3}$$	Jerk (physics)/Rate of acceleration
n-th	$$f^{(n)}(x)$$	$$\frac{d^n y}{dx^n}$$	n-th order phenomena

For implicit differentiation: Derivatives up to $$d^n y/dx^n$$ can be calculated if needed.

📋 How to Calculate Higher Derivatives

Find the first derivative $$f'(x)$$ using your usual rules.
Differentiating again gets the second derivative $$f''(x)$$.
Keep differentiating: each time you get the next order (third, fourth, etc...).
Simplify at every step for easier calculation.
For products and quotients, apply their rules every time you repeat differentiation.

Shortcuts: For $$e^{kx}$$, trigs, and polynomials, see patterns below!

🧮 Examples of Higher-Order Derivatives

Example 1: Polynomial

$$f(x)=x^4-2x^2+3x$$
$$f'(x)=4x^3-4x+3$$
$$f''(x)=12x^2-4$$
$$f'''(x)=24x$$
$$f^{(4)}(x)=24$$ (all higher-order derivatives vanish)

Example 2: Trigonometric Function

$$f(x)=\sin x$$
$$f'(x)=\cos x$$
$$f''(x)=-\sin x$$
$$f'''(x)=-\cos x$$
$$f^{(4)}(x)=\sin x$$ (cycles every 4)

Example 3: Exponential Function

$$g(x)=e^{kx}$$
$$g'(x)=ke^{kx}$$
$$g''(x)=k^2e^{kx}$$
$$g^{(n)}(x)=k^n e^{kx}$$

Example 4: Logarithmic Function

$$y=\ln x$$
$$y' = \frac{1}{x}$$
$$y'' = -\frac{1}{x^2}$$
$$y''' = \frac{2}{x^3}$$
$$y^{(n)} = \frac{(-1)^{n-1}(n-1)!}{x^n}$$

💡 Patterns, Shortcuts, and Tricks

Polynomials: Keep differentiating until you run out of powers—all higher-order terms after degree n become 0.
Trig functions: Cycle every 4 derivatives: $$\sin \to \cos \to -\sin \to -\cos$$.
Exponentials: $$e^{kx}$$ higher derivatives just pick up powers of $$k$$.
Logarithms: Each differentiation multiplies by -1 and increases the denominator power.
After each differentiation, simplify before differentiating again. It reduces errors!
Apply product/quotient rules at every level if functions are multiplied/divided.
For implicit derivatives, each order is obtained by differentiating the previous equation.

❌ Common Mistakes

Forgetting to simplify derivatives before proceeding.
Mixing up notation between powers and derivative order.
Missing patterns/cycles in trigonometric or exponential derivatives.
Neglecting the repeated use of chain, product, or quotient rules for higher-order derivatives of complex functions.

📝 Practice Problems

Try These Yourself:

Find $$f''(x)$$ and $$f^{(3)}(x)$$ for $$f(x)=x^6-2x^3+5$$
Find $$g^{(4)}(x)$$ for $$g(x)=e^{5x}$$
Find $$h''(x)$$ for $$h(x)=x^2\sin x$$ (product rule is needed)
Find $$y'''$$ for $$y=\ln x$$

Answers:

$$f''(x)=30x^4-12x$$; $$f^{(3)}(x)=120x^3-12$$
$$g^{(4)}(x)=625e^{5x}$$
First: $$h'(x)=2x\sin x + x^2\cos x$$;
Second: $$h''(x)=2\sin x + 2x\cos x + 2x\cos x + x^2(-\sin x)=2\sin x + 4x\cos x - x^2\sin x$$
$$y'''=\frac{2}{x^3}$$

✏️ AP® Exam Tips – Higher-Order Derivatives

Write clear notation for derivative order: use primes, powers, or Leibniz as required.
Simplify after each order before proceeding; it saves time and errors.
For trigonometric and exponential functions, look for cycles/shortcuts.
Box your final answer—especially in FRQs—use correct notation.
Practice with mixed rules (product, quotient, chain!) for composite/higher-order derivatives.