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Higher Order Derivative Solver

Compute 1st through 5th order derivatives step by step

Select Function Type
Order of derivative: up to n =
f(x)=x^

Higher Order Derivatives

f = first derivative (slope)
f = second derivative (concavity)
f = third derivative (jerk)
f^(4) = fourth derivative (etc.)

Higher order derivatives are obtained by repeatedly differentiating a function. Each application of the derivative operator reduces polynomial degree by 1. Trigonometric functions cycle through patterns. Exponential functions always keep their form.

⚠ The more times you differentiate, the simpler the result often becomes. Polynomials eventually reach 0. Sine/cosine cycle forever.

What Are Higher Order Derivatives?

The nth derivative is found by differentiating n times. For polynomials: each differentiation reduces the degree by 1. A degree n polynomial has constant nth derivative and zero (n+1)th derivative. For e^x: all derivatives equal e^x. For sin/cos: cycles every 4 derivatives.

Polynomial Rule

d^n/dx^n x^k = k!/(k-n)! * x^(k-n) for k>=n. For k=n: n! (constant). For k

Trig Cycle

sin -> cos -> -sin -> -cos -> sin (4-step cycle). f^(n)(sin) = sin(x+n*pi/2). cos(x) follows similar pattern.

Exponential

d^n/dx^n e^(ax) = a^n * e^(ax). The exponential function is its own derivative up to the factor a^n.

Logarithm

d/dx ln(x)=1/x. d^2/dx^2 ln(x)=-1/x^2. d^3/dx^3 ln(x)=2/x^3. nth derivative: (-1)^(n-1)*(n-1)!/x^n.

Teaching Example: f(x)=3x^3. f=9x^2, f=18x, f=18, f^(4)=0. Degree 3 polynomial: 3rd derivative is constant, 4th is zero.

Applications

Concavity Physics Taylor Series Optimization Differential Eq

Frequently Asked Questions

What is higher order derivative?▼
Derivative of a derivative. 2nd = concavity, 3rd = jerk. Differentiate repeatedly to get higher orders.
Polynomial nth derivative?▼
Each differentiation reduces degree by 1. Degree n polynomial: nth derivative = constant * n!, (n+1)th = 0.
Sin x 4th derivative?▼
f^(4)(sin)=sin(x) after 4 differentiations. Cycle: sin->cos->-sin->-cos->sin.
e^x nth derivative?▼
All derivatives of e^x equal e^x. For e^(ax): nth derivative = a^n * e^(ax). Never becomes zero.

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