Differentiation Rules

Differentiation Rules

1. Product Rule

   • Use: when you have to functions of x.
     • Formula: y'= f(x) g'(x) + g(x) f'(x)

2. Power Rule

   • Use: when you have only a coefficient multiplying a variable with an exponent.
     • Formula: y=x^n, then y'=nx^n-1

3. Chain Rule

   • Use: when you have 2 terms to a power
     • Formula: y=U^n, y'=n (U^n-1) (U')

4. Quotient Rule

   • Use: If 2 functions are diving each other
     • Formula: g(x) f'(x) - g'(x) f(x)/ (g(x))^2

5. Exponential Function (e)

   • Use: when you have "e" as a base.
     • Formula: 1) If y=e^x, then y'= e^x
       • 2) If y=e^u, then y'=u' (e^u)

6. Derivatives of Natural Logarithms

   • Use: when you have a natural logarithm involved (ln)
     • Formula: 1) y= ln(x) --> y'= 1/x
       • 2) y= ln(u) --> y'= u'/u

7. ln Rules

   • Use: when you have a 2 variables multiplying or dividing with ln.
     • Formula: 1) ln(ab)= ln(a) + ln(b)
       • 2) ln(a/b)= ln(a) - ln(b)
         • 3) ln(a^n)= (n)ln(a)

8. Trigonometric Functions

   • Use: when you need to find the derivative of a trigonometric function.
     • Formula: y=sin(x) --> y'=cos(x)
       • y=sin(u) --> y'= u' cos(u)
         • y= cos(x) --> y'= -sin(x)
           • y= cos(u) --> y'=-u' sin(u)

9. Trigonometric Identities

   • Use: to simplify trigonometric functions
     • Formula: Tan(x)= sin(x)/ cos(x)
       • Cot(x)= cos(x)/ sin(x) or 1/tan(x)
         • Csc(x)=1/sin(x)
           • Sec(x)= 1/cos(x)
             • Sin^2(x) + cos^2(x)= 1
               • 1 + tan^2 (x)= sec^2 (x)
                 • 1 + cot^2 (x)= csc^2 (x)
   • Derivatives:
     • f(x)= sin (u), then f'(x)= u' cos (u)
       • If f(x)= cos (u), then f'(x)= -u' sin(u)
     • If f(x)= tan(u), then f'(x)= u' sec^2 (u)
       • If f(x)= cot(u), then f'(x)= -u' csc^2(u)
         • If f(x)= sec (u), then f'(x)= u' sec (u) tan (u)
           • If f(x)= csc(u), then f'(x)= -u csc (u) cot (u)

10. Higher Order Derivaties

    • Use: different notations of a higher number of derivatives, more than just the first one.
      • 1st derivative: y'= f'(x)= dy/dx = df(x)/dx
        • 2nd derivative: y''= f''(x)= d^2y/dx^2= d^2f(x)/dx^2
          • 3rd derivative: y'''=f'''(x)= d^3y/dx^3= d^3f(x)/dx^3
            • nth derivative: y^n=f^(n)x= d^ny/dx^n

11. Implicit Differentiation

    • Use: when you have the derivative of "y"
      • Formula: multiply every "y" times dy/dx, then solve it for this.

12. Related Rates

    • Use: when instead of solving it in respect of "x", you solve it in respect of "y".
      • Formula: If y= f(u) and u= y(x)
        • dy/dx= (dy/du) (du/dx)