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3.6 Calculating Higher-Order Derivatives

Syllabus
2020
Topic
3.6
Level
—

Learning objectives

—FUN-3.F—Determine higher order derivatives of a function• FUN-3.F Determine higher order derivatives of a function.• FUN-3.F.1 Differentiating f′ produces the second derivative f″, provided the derivative of f′ exists; repeating this process produces higher-order derivatives of f.• FUN-3.F.2 Higher-order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include d²y/dx², f″(x), and y″. Higher-order derivatives can be denoted dⁿy/dxⁿ or f⁽ⁿ⁾(x).• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

Differentiate Again for Higher Orders

A higher-order derivative is obtained by differentiating the previous derivative. The second derivative is the derivative of the first derivative; repeating the process produces the third, fourth, and nth derivatives whenever each required derivative exists.

f''(x)=y''=\frac{d^2y}{dx^2},\qquad f^{(n)}(x)=\frac{d^ny}{dx^n}

  1. Differentiate the original function to obtain f′f'.\n2. Treat f′f' as the new function and differentiate it to obtain f′′f''.\n3. Continue one order at a time until the requested derivative is reached.\n4. Label each result with its order so that no stage is skipped.

For f(x)=x4−2x3+5f(x)=x^4-2x^3+5, f′(x)=4x3−6x2,f'(x)=4x^3-6x^2, f′′(x)=12x2−12x,f''(x)=12x^2-12x, and f′′′(x)=24x−12.f'''(x)=24x-12. Each line is found by differentiating the line immediately before it.

d2y/dx2d^2y/dx^2 means differentiate twice; it is not (dy/dx)2(dy/dx)^2. Also, a requested higher derivative exists only if the preceding derivative is differentiable at the point or throughout the interval being considered.