Derivatives

1. Product Rule

1.1. 3 tanx + 6sinx = 3 (csc^2 x) + 6 cosx 3 csc^2x + 6 cosx

2. 3 tanx + 6sinx = 3 (csc^2 x) + 6 cosx 3 csc^2x + 6 cosx

3. g'(x)= 3 sec(x) - 10 cot(x) g'(x)= 3 secx tanx - 10(-csc^2 x) g'(x)= 3 secx tanx + 10 csc^2 x

4. Power Rule

4.1. x^6 + 2x^3 = 1 x 6x^6-1 + 2 x 3x^3-1 = 6x^5 + 6x^2

5. x^6 + 2x^3 = 1 x 6x^6-1 + 2 x 3x^3-1 = 6x^5 + 6x^2

6. Higher order Derivatives

6.1. f(x)= x^3 - 6x^5 = f'(x)= 3x^2 - 30x^4 = f''(x)= 6x - 120X^3 f'''(x)= 6 - 360^2 =f''''(x)= -720x f^5(x)= -720 f^6(x) = 0 = f^n(x)= 0 f^100(x)= 0

7. f(x)= x^3 - 6x^5 = f'(x)= 3x^2 - 30x^4 = f''(x)= 6x - 120X^3 f'''(x)= 6 - 360^2 =f''''(x)= -720x f^5(x)= -720 f^6(x) = 0 = f^n(x)= 0 f^100(x)= 0

8. Derivative of Trigonometric function

9. Quotient Rule

9.1. h(x)= x^3 - 4 / x^2 + 1 h'(x)= f'(x)g(x) - f(x)g'(x) =3x^2 (x^2 + 1) - (x^3 - 4) (2x) / (x^2 +1)^2 = x^4 + 3x^2 + 8x / (x^2 + 1)

10. h(x)= x^3 - 4 / x^2 + 1 h'(x)= f'(x)g(x) - f(x)g'(x) =3x^2 (x^2 + 1) - (x^3 - 4) (2x) / (x^2 +1)^2 = x^4 + 3x^2 + 8x / (x^2 + 1)