Differentiating of algebraic functions

1. Derivatives of Polynomial Functions:

  • Constant Function:

    • ddx[c]=0 where c is a constant.
  • Linear Function:

    • ddx[ax+b]=a where a and b are constants.
  • Quadratic Function:

    • ddx[ax2+bx+c]=2ax+b

2. Derivatives of Exponential and Logarithmic Functions:

  • Exponential Function:

    • ddx[ex]=ex
    • ddx[ax]=axln(a) where a>0
  • Natural Logarithm Function:

    • ddx[ln(x)]=1x

3. Derivatives of Radical Functions:

  • Square Root Function:

    • ddx[x]=12x

 

  • Higher Order Roots:

    • ddx[xn]=nxn1 where n is a constant.

4. Derivatives of Rational Functions:

  • Constant Divided by a Function:

    • ddx[cf(x)]=cf(x)[f(x)]2
  • Quotient Rule for Functions:

    • ddx[f(x)g(x)]=g(x)f(x)f(x)g(x)[g(x)]2

5. Chain Rule for Algebraic Functions:

  • Chain Rule Application:
    • When functions are nested or composed, the chain rule is applied.
    • Example: f(x)=e3x2
      • f(x)=6xe3x2