Basic Differentiation

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Reminders

Notation:

Function First Derivative Second Derivative

f (x) f ' (x) f '' (x)

g(x) g ' (x) g '' (x)

y y ' or (dy) / (dx) y '' or (d^2)(y) / (d)(x^2)

The Power Rule:

If y = x^n , then (dy) / (dx) = nx ^ (n-1)

If y = x , then (dy) / (dx) = 1

If y = kx , then (dy) / (dx) = k (where k is a constant)

If y = k , then (dy) / (dx) = 0 (where k is a constant)

The Addition Rule:

If y = ax^n + bx^m , where a and b are constants, then (dy) / (dx) = a ( nx ^ (n-1) ) + b ( mx ^ (m-1) )

If y = k / x , then (dy) / (dx) = - k / x^2

If y = k radical x , then (dy) / (dx) = k / 2 radical x

The Product Rule:

If f(x) = uv , then f ' (x) = u ( (dv) / (dx) ) + v ( (du) / (dx) )

The Quotient Rule:

If f(x) = u / v , then f ' (x) = [ (v ( du / dx ) ) - ( u ( dv / dx ) ) ] / v^2

The Chain Rule:

If y = f ( g ( x ) ) , then y ' = [ ( df ( g ( x ) ) ) / ( dg ) ] [ dg / dx ]