Methods of Differentiation

 

Methods of Differentiation: A Quick Reference

Differentiation is a mathematical process used to find the rate of change of a function. It's a fundamental concept in calculus. Here are some common methods used for differentiation:  

1. Power Rule:

• For functions of the form f(x) = x^n, where n is any real number except -1:
  • f'(x) = nx^(n-1)

2. Sum and Difference Rule:

• If f(x) = u(x) ± v(x), then:
  • f'(x) = u'(x) ± v'(x)

3. Product Rule:

• If f(x) = u(x) * v(x), then:
  • f'(x) = u'(x) * v(x) + u(x) * v'(x)

4. Quotient Rule:

• If f(x) = u(x) / v(x), then:
  • f'(x) = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]^2

5. Chain Rule:

• If f(x) = g(h(x)), then:
  • f'(x) = g'(h(x)) * h'(x)

6. Implicit Differentiation:

• For equations that cannot be easily solved for y, differentiate both sides with respect to x, treating y as a function of x.

7. Logarithmic Differentiation:

• For functions of the form f(x) = [g(x)]^h(x), take the natural logarithm of both sides and then differentiate.

8. Trigonometric Functions:

• The derivatives of common trigonometric functions are:
  • d/dx(sin(x)) = cos(x)  
  • d/dx(cos(x)) = -sin(x)  
  • d/dx(tan(x)) = sec^2(x)  
  • d/dx(cot(x)) = -csc^2(x)  
  • d/dx(sec(x)) = sec(x)tan(x)  
  • d/dx(csc(x)) = -csc(x)cot(x)  

9. Inverse Trigonometric Functions:

• The derivatives of common inverse trigonometric functions are:
  • d/dx(arcsin(x)) = 1 / √(1 - x^2)  
  • d/dx(arccos(x)) = -1 / √(1 - x^2)  
  • d/dx(arctan(x)) = 1 / (1 + x^2)  
  • d/dx(arccot(x)) = -1 / (1 + x^2)  
  • d/dx(arcsec(x)) = 1 / (|x|√(x^2 - 1))  
  • d/dx(arccsc(x)) = -1 / (|x|√(x^2 - 1))  

By understanding these methods, you can effectively differentiate a wide range of functions and solve calculus problems.