Mathfolis

Higher-Order Derivatives

Unit 3 · Differentiation — Composite, Implicit, Inverse

Higher-Order Derivatives

The second derivative f″(x) is the derivative of f′(x), and measures concavity. The nth derivative is obtained by differentiating n times. Physically: if s(t) is position, then s′(t) = velocity and s″(t) = acceleration.

Notation

f′′(x)=d2ydx2f′′′(x)=d3ydx3f(n)(x)=dnydxnf''(x) = \frac{d^2y}{dx^2} \qquad f'''(x) = \frac{d^3y}{dx^3} \qquad f^{(n)}(x) = \frac{d^ny}{dx^n}

Multi-Layer Chain Rule

Example pattern
ddx[f(g(h(x)))]=f′(g(h(x)))⋅g′(h(x))⋅h′(x)\frac{d}{dx}[f(g(h(x)))] = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)
AP Tip: For sin x, the derivatives cycle with period 4: cos x → −sin x → −cos x → sin x → … Use this to find any nth derivative quickly.
Type 1

Higher-Order Derivatives of Polynomials

Differentiate repeatedly using the power rule. For polynomials, the nth derivative is eventually zero.

Example 1
Find f″(x) and f‴(x).
f(x)=2x5−x3+4xf(x) = 2x^5 - x^3 + 4x

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Type 2

Multi-Layer Chain Rule

Work from the outside in: identify the outermost function, differentiate it, then multiply by the derivative of the inner function(s).

Example 2
Find dy/dx.
y=sin⁡4(3x)y = \sin^4(3x)
Example 3
Find dy/dx.
y=e−2x2y = e^{-2x^2}

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Type 3

Second Derivative via Implicit Differentiation

Find dy/dx first, then differentiate again implicitly. Substitute dy/dx back to express d²y/dx² in terms of x and y only.

Example 4
Find d²y/dx² for the curve in terms of x and y.
y2=x3y^2 = x^3

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