Compute 1st through 5th order derivatives step by step
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 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.
d^n/dx^n x^k = k!/(k-n)! * x^(k-n) for k>=n. For k=n: n! (constant). For k
sin -> cos -> -sin -> -cos -> sin (4-step cycle). f^(n)(sin) = sin(x+n*pi/2). cos(x) follows similar pattern.
d^n/dx^n e^(ax) = a^n * e^(ax). The exponential function is its own derivative up to the factor a^n.
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.
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