Fundamental Rules for Differentiation

1. Power Rule:

Rule:

  • For a function f(x)=xn, where n is a constant, the derivative is f(x)=nxn1.

Examples:

  • f(x)=x3    f(x)=3x2
  • g(x)=x2    g(x)=2x3

2. Constant Multiple Rule:

Rule:

  • If c is a constant and f(x) is differentiable, then ddx[cf(x)]=cddx[f(x)].

Example:

  • f(x)=5x2    f(x)=52x=10x

3. Sum/Difference Rule:

Rule:

  • The derivative of the sum/difference of two functions is the sum/difference of their derivatives: ddx[f(x)±g(x)]=ddx[f(x)]±ddx[g(x)].

Example:

  • f(x)=x2+3x    f(x)=2x+3

4. Product Rule:

Rule:

  • For two functions u(x) and v(x), the derivative of their product is: ddx[u(x)v(x)]=u(x)v(x)+u(x)v(x).

Example:

  • f(x)=x2sin(x)    f(x)=2xsin(x)+x2cos(x)

5. Quotient Rule:

Rule:

  • For u(x) and v(x) where v(x)0, the derivative of their quotient is: ddx[u(x)v(x)]=v(x)u(x)u(x)v(x)[v(x)]2.

Example:

  • f(x)=x2sin(x)    f(x)=sin(x)2xx2cos(x)[sin(x)]2

6. Chain Rule:

Rule:

  • If y=f(u) and u=g(x), then the chain rule states that dydx=dydududx.

Example:

  • f(x)=sin(x2)    f(x)=2xcos(x2)