Differentiation of a Basic Function

Differentiation of a Basic Function

Basic Derivatives

• The derivative of a constant c is 0.
• The derivative of x^n is n*x^(n-1), where n is any real number.
• The derivatives of sin(x), cos(x), and tan(x) are cos(x), -sin(x), and sec^2(x) respectively.

Derivatives of Sum and Differences

• The derivative of a sum or difference of two or more functions is the sum or difference of their derivatives, respectivley.

Product and Quotient Rule

• The product rule: the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.
• The quotient rule: the derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all over the bottom function squared.

Chain Rule

• If you have a function composed of two other functions, the chain rule is used to differentiate it. The derivative of the composite function is the derivative of the outer function times the derivative of the inner function.

Derivatives in Real-World Applications

• Differentiation has many real-world applications such as calculating rates of change, determining maximum and minimum values, and solving problems involving motion.

Remember, practice is essential for the mastery of differentiation, so be sure to work through multiple problems for each rule for best understanding.