Calculating
Higher-Order Derivatives
Master the art of differentiating repeatedly to uncover deeper behavior of functions: concavity, inflection points, and beyond.
AP Calculus AB
What Is a Higher-Order Derivative?
When you take the derivative of a function, you get its first derivative \( f'(x) \). But you can differentiate again: the derivative of \( f'(x) \) is the second derivative \( f''(x) \), and the derivative of \( f''(x) \) is the third derivative \( f'''(x) \), and so on.
A continuous route through hills: elevation, grade, and intermediate height
For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.Height, grade and changing grade
For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.Notation for Higher-Order Derivatives
Lagrange Notation
\( f'(x),\; f''(x),\; f'''(x),\; f^{(n)}(x) \)
Leibniz Notation
\( \dfrac{dy}{dx},\; \dfrac{d^2y}{dx^2},\; \dfrac{d^3y}{dx^3},\; \dfrac{d^ny}{dx^n} \)
D-Operator Notation
\( Dy,\; D^2y,\; D^3y,\; D^ny \)
Why Do They Matter?
\( f'(x) \): Rate of Change
Velocity, slope of the tangent line, increasing/decreasing behavior.
\( f''(x) \): Concavity
Acceleration, concave up vs. down, inflection points.
\( f'''(x) \) and Beyond
Jerk (rate of acceleration change), Taylor series, and physics applications.
The Core Principle
To find the \( n \)-th derivative, apply the derivative operator \( n \) times in succession:
All standard differentiation rules apply at each step: power rule, product rule, chain rule, and quotient rule.
Power Rule Pattern
For \( f(x) = x^n \), repeated differentiation follows a clean pattern:
\( f'(x) \) vs \( f''(x) \): Key Distinction
| \( f'(x) \): First Derivative | \( f''(x) \): Second Derivative | |
|---|---|---|
| Measures | Rate of change (slope) | Rate of change of the slope |
| Sign tells us | Increasing (\( > 0 \)) or decreasing (\( < 0 \)) | Concave up (\( > 0 \)) or concave down (\( < 0 \)) |
| Zeros indicate | Critical points | Possible inflection points |
| Physics analogy | Velocity | Acceleration |
Visualizing \( f \), \( f' \), and \( f'' \)
How to Calculate Higher-Order Derivatives
1
Find \( f'(x) \). Apply the appropriate rule (power, product, chain, quotient) to get the first derivative.
2
Simplify fully. Combine like terms, factor if possible: this makes the next differentiation easier.
3
Differentiate \( f'(x) \) to get \( f''(x) \). Treat \( f'(x) \) as a new function and repeat all derivative rules.
4
Repeat for higher orders. Continue differentiating: \( f'' \to f''' \to f^{(4)} \), simplifying at each step.
5
Look for patterns. Many functions (polynomials, exponentials, trig) produce predictable patterns after a few derivatives.
Example 1: Polynomial
Find the first three derivatives of \( f(x) = 2x^4 - 5x^3 + x^2 - 7x + 3 \).
1
\( f'(x) \): Apply the power rule term by term: \( f'(x) = 8x^3 - 15x^2 + 2x - 7 \)
2
\( f''(x) \): Differentiate \( f'(x) \): \( f''(x) = 24x^2 - 30x + 2 \)
3
\( f'''(x) \): Differentiate \( f''(x) \): \( f'''(x) = 48x - 30 \)
Example 2: Trigonometric Cycle
Find the first four derivatives of \( f(x) = \sin(x) \).
Example 3: Exponential Function
Find all derivatives of \( f(x) = e^{3x} \).
1
\( f'(x) \): By the chain rule: \( f'(x) = 3e^{3x} \)
2
\( f''(x) \): Differentiate again: \( f''(x) = 9e^{3x} = 3^2 e^{3x} \)
3
\( f'''(x) \): And again: \( f'''(x) = 27e^{3x} = 3^3 e^{3x} \)
Example 4: When the Product Rule Is Needed
Find \( f''(x) \) for \( f(x) = x^2 \sin(x) \).
1
First derivative (product rule): \( f'(x) = 2x\sin(x) + x^2\cos(x) \)
2
Second derivative: differentiate each term using the product rule again:
\( \frac{d}{dx}[2x\sin(x)] = 2\sin(x) + 2x\cos(x) \)
\( \frac{d}{dx}[x^2\cos(x)] = 2x\cos(x) - x^2\sin(x) \)
\( \frac{d}{dx}[2x\sin(x)] = 2\sin(x) + 2x\cos(x) \)
\( \frac{d}{dx}[x^2\cos(x)] = 2x\cos(x) - x^2\sin(x) \)
3
Combine: \( f''(x) = 2\sin(x) + 4x\cos(x) - x^2\sin(x) \)
Common Mistakes to Avoid
✕ Forgetting the Chain Rule
When differentiating \( e^{3x} \), students often write \( e^{3x} \) instead of \( 3e^{3x} \). The inner derivative multiplies every time.
✕ Skipping Simplification
Carrying unsimplified expressions forward makes each successive derivative exponentially harder and error-prone.
✕ Wrong Leibniz Notation
Writing \( \frac{d^2y}{d^2x} \) or \( \frac{dy^2}{dx^2} \) instead of the correct \( \frac{d^2y}{dx^2} \). The exponent goes on the \( d \), not \( x \) or \( y \).
Summary of Key Patterns
| Function | \( f^{(n)}(x) \) | Pattern |
|---|---|---|
| \( x^k \) | \( \frac{k!}{(k-n)!}\,x^{k-n} \) | Becomes 0 for \( n > k \) |
| \( e^{ax} \) | \( a^n e^{ax} \) | Never zero |
| \( \sin(x) \) | \( \sin\!\left(x + \frac{n\pi}{2}\right) \) | Cycles with period 4 |
| \( \cos(x) \) | \( \cos\!\left(x + \frac{n\pi}{2}\right) \) | Cycles with period 4 |
| \( \ln(x) \) | \( \frac{(-1)^{n-1}(n-1)!}{x^n} \) | Never zero for \( x > 0 \) |
Question 1 of 5
If \( f(x) = 3x^5 - 2x^3 + x \), what is \( f''(x) \)?
First, \( f'(x) = 15x^4 - 6x^2 + 1 \). Then \( f''(x) = 60x^3 - 12x \). Choice (a) is the first derivative, (c) forgot to drop the constant, and (d) is the third derivative.
Question 2 of 5
What is the 50th derivative of \( \cos(x) \)?
The derivatives of \( \cos(x) \) cycle with period 4: \( \cos, -\sin, -\cos, \sin, \cos, \ldots \) Since \( 50 = 4 \times 12 + 2 \), the 50th derivative is the same as the 2nd: \( -\cos(x) \).
Question 3 of 5
If \( g(x) = e^{-2x} \), what is \( g'''(x) \)?
Using the pattern \( f^{(n)}(x) = a^n e^{ax} \) with \( a = -2 \): \( g'''(x) = (-2)^3 e^{-2x} = -8e^{-2x} \).
Question 4 of 5
Find \( \dfrac{d^2y}{dx^2} \) if \( y = x^3 \ln(x) \).
First: \( y' = 3x^2\ln(x) + x^3 \cdot \frac{1}{x} = 3x^2\ln(x) + x^2 \). Then: \( y'' = 6x\ln(x) + 3x^2 \cdot \frac{1}{x} + 2x = 6x\ln(x) + 3x + 2x = 6x\ln(x) + 5x \). Choice (a) is the first derivative. Choices (b) and (c) have arithmetic errors.
Question 5 of 5
If \( f(x) = x^6 \), what is \( f^{(7)}(x) \)?
For \( x^6 \), the 6th derivative is \( 6! = 720 \) (a constant). The 7th derivative of a constant is 0. Choice (a) is \( f^{(6)}(x) \) and (d) is \( 7! \), which is unrelated.