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\( f'' \)
\( f''' \)

Calculating
Higher-Order Derivatives

Master the art of differentiating repeatedly: from the second derivative \( f''(x) \) to the \( n \)th derivative \( f^{(n)}(x) \), and discover how each layer reveals deeper behavior of functions.

AP Calculus BC

What Are Higher-Order Derivatives?

When you differentiate a function \( f(x) \), you get the first derivative \( f'(x) \). But you can keep going: differentiate the first derivative to get the second derivative \( f''(x) \), then differentiate again for the third derivative \( f'''(x) \), and so on.

Notation for Higher-Order Derivatives

Lagrange (Prime)
\( f'(x),\; f''(x),\; f'''(x),\; f^{(4)}(x) \)
Leibniz (Fraction)
\( \dfrac{dy}{dx},\; \dfrac{d^2 y}{dx^2},\; \dfrac{d^3 y}{dx^3},\; \dfrac{d^n y}{dx^n} \)
Euler (Operator)
\( Df,\; D^2 f,\; D^3 f,\; D^n f \)

The Derivative Chain Visualized

\( f(x) \)
\( f'(x) \)
\( f''(x) \)
\( f'''(x) \)
\( \frac{d}{dx} \)
\( \frac{d}{dx} \)
\( \frac{d}{dx} \)
 Position Velocity Acceleration Jerk

In physics, each successive derivative reveals a new layer of motion: position \(\to\) velocity \(\to\) acceleration \(\to\) jerk. The same principle applies to any function.

Why Do They Matter?

1
Concavity and inflection points: \( f''(x) \) tells us whether a curve bends upward or downward.
2
Second Derivative Test: Classify critical points as local maxima or minima using the sign of \( f''(c) \).
3
Taylor and Maclaurin series: Higher-order derivatives are the building blocks for polynomial approximations of functions.
4
Differential equations: Many physical laws involve second-order (or higher) derivatives, such as \( F = ma = m\,f''(t) \).

Core Rule: Polynomial Derivatives

Each differentiation reduces the exponent by 1 and multiplies by the current exponent. After \( n \) steps, a degree-\( n \) polynomial becomes a constant, and any further derivative is zero.

Key Formulas: Exponentials and Trig

Exponential
\( \dfrac{d^n}{dx^n}\big[e^x\big] = e^x \) for all \( n \). The exponential is its own derivative at every order.
Sine Cycle
\( \sin x \to \cos x \to -\sin x \to -\cos x \to \sin x \). Period: 4 derivatives.
Cosine Cycle
\( \cos x \to -\sin x \to -\cos x \to \sin x \to \cos x \). Same period of 4.

The Sine Derivative Cycle

\( \sin x \)
\( \cos x \)
\( -\sin x \)
\( -\cos x \)
\( \frac{d}{dx} \)
\( \frac{d}{dx} \)
\( \frac{d}{dx} \)
\( \frac{d}{dx} \)

Differentiating \( \sin x \) cycles through four functions and returns to the start. This cycle repeats forever: the 100th derivative of \( \sin x \) equals \( \sin x \) because \( 100 \bmod 4 = 0 \).

Terminating vs. Cyclic vs. Persistent

Behavior Terminating Cyclic Persistent
Pattern Eventually reaches 0 Repeats in a cycle Never zero, never repeats
Example \( x^5 \) \( \sin x,\; \cos x \) \( e^x,\; \ln x \)
\( f^{(n)} \) for large \( n \) 0 Cycles (period 4) Always nonzero
Key insight Degree drops each step Use \( n \bmod 4 \) Apply pattern formulas

How to Calculate Higher-Order Derivatives

1
Identify the function type: Is it polynomial, exponential, trigonometric, or a combination?
2
Compute the first derivative using standard rules (power, chain, product, quotient).
3
Differentiate again to get \( f''(x) \). Continue iterating for higher orders.
4
Look for a pattern: If asking for \( f^{(n)} \), compute the first few derivatives and identify a formula or cycle.
5
Apply the pattern to jump directly to the \( n \)th derivative without computing every intermediate step.

Example 1: Polynomial

1
\( f'(x) = 10x^4 - 9x^2 + 7 \)
2
\( f''(x) = 40x^3 - 18x \)
3
\( f'''(x) = 120x^2 - 18 \)

Example 2: Trig with Chain Rule

1
First derivative: \( f'(x) = 3\cos(3x) \) by the chain rule.
2
Second derivative: \( f''(x) = -9\sin(3x) \) by the chain rule again.

Example 3: Product Rule Iterated

1
First derivative (product rule): \( f'(x) = 2x\,e^x + x^2 e^x = e^x(x^2 + 2x) \)
2
Second derivative (product rule again):
\( f''(x) = e^x(x^2 + 2x) + e^x(2x + 2) = e^x(x^2 + 4x + 2) \)

Pattern: \( e^x \) Times a Polynomial

When \( f(x) = e^x \cdot p(x) \) where \( p(x) \) is a polynomial, each derivative has the form \( e^x \cdot q(x) \) for some polynomial \( q \) of the same degree.

\( f = e^x x^2 \)
\( f' = e^x \)
\( (x^2+2x) \)
\( f'' = e^x \)
\( (x^2+4x+2) \)
 ... degree 2 degree 2 degree 2

Common Mistakes to Avoid

✕ Forgetting the Chain Rule
When differentiating \( \sin(3x) \), each derivative must multiply by 3. Missing this gives wrong coefficients: \( f''(x) \neq -\sin(3x) \).
✕ Wrong Trig Cycle Position
Miscounting in the \( \sin/\cos \) cycle. Always compute \( n \bmod 4 \) carefully: remainder 2 means negative of the original, not positive.
✕ Sign Errors in Products
Forgetting negative signs when differentiating products involving \( \cos x \to -\sin x \). Track each sign change explicitly.

Summary of Key Formulas

Function \( n \)th Derivative Notes
\( x^k \) \( \frac{k!}{(k-n)!}\,x^{k-n} \) Zero when \( n > k \)
\( e^x \) \( e^x \) Always itself
\( e^{ax} \) \( a^n e^{ax} \) Chain rule factor
\( \sin(x) \) \( \sin\!\big(x + \frac{n\pi}{2}\big) \) Period-4 cycle
\( \cos(x) \) \( \cos\!\big(x + \frac{n\pi}{2}\big) \) Period-4 cycle
\( \ln(x) \) \( \frac{(-1)^{n-1}(n-1)!}{x^n} \) Never zero

Question 1 of 5

What is the third derivative of \( f(x) = 5x^4 \)?

Question 2 of 5

What is \( \dfrac{d^{50}}{dx^{50}}[\cos x] \)?

Question 3 of 5

If \( f(x) = e^{2x} \), what is \( f^{(4)}(x) \)?

Question 4 of 5

Find \( f''(x) \) if \( f(x) = x \cdot \ln x \).

Question 5 of 5

What is the second derivative of \( f(x) = \sin(5x) \)?