Higher-order derivatives are obtained by repeated differentiation. The second derivative f''(x) measures the rate of change of the slope (concavity). The nth derivative follows patterns for standard functions.

Recognizable Patterns:

• (e^(ax))^(n) = $a^n$ * e^(ax) — exponential reproduces itself
• (sin(ax+b))^(n) = $a^n$ * sin(ax + b + n*pi/2) — each derivative shifts phase by pi/2
• (cos(ax+b))^(n) = $a^n$ * cos(ax + b + n*pi/2) — same phase shift
• ($x^m$)^(n) = m!/(m-n)! * x^(m-n) for n <= m; 0 for n > m
• (ln x)^(n) = (-1)^(n-1) * (n-1)! / $x^n$

Leibniz's Theorem generalizes the product rule to nth derivatives: (uv)^(n) = sum(r=0 to n) C(n,r) * u^(n-r) * v^(r). This is particularly useful when one factor is a polynomial (so its higher derivatives vanish, truncating the sum).

JEE Applications:

1. Finding y^(n) for products like $x^2$*$e^x$ (Leibniz with polynomial factor)
2. Proving differential equations: if y = e^(a*sin^(-1)(x)), show (1-$x^2$)y'' - xy' = $a^2$*y
3. Finding y'' for parametric curves using the correct parametric second derivative formula

Connection to Taylor Series: The Taylor expansion of f(x) around x = a uses all derivatives: f(x) = sum f^(n)(a)*(x-a)^n/n!. So finding patterns in higher derivatives directly connects to series expansions.