Basic Differentiation

Basic Differentiation

Definition and Fundamental Theorem

  • Differentiation is a fundamental operation in calculus that produces a function showing the rate of change of one quantity compared to another.
  • The principal concept behind differentiation is the notion of the limit.
  • The Fundamental Theorem of Calculusstates that differentiation and integration are inverse processes.

Power Rule

  • The Power Rule for differentiation states that if y = x^n, its derivative is given by dy/dx = n*x^(n-1).
  • This is applicable to any real number n (n ≠ -1).

Constant, Sum and Difference Rule

  • The derivative of a constant is zero.
  • The derivative of the sum or difference between two functions is the sum or difference of their derivatives.

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 derivative of the bottom function, all over the bottom function squared.

Chain Rule

  • The Chain Rule: When differentiating composite functions, the derivative of the outer function is multiplied by the derivative of the inner function.

Implicit Differentiation

  • Implicit differentiation is used when it is difficult to solve an equation for y explicitly as a function of x.
  • The technique is to differentiate each side of the equation with respect to x, treating y as a function of x, and consequently applying the chain rule when necessary.

Higher Derivatives

  • Second or higher derivatives can be found by differentiating the derivative of a function.
  • The second derivative often represents information about the geometry of the function, for instance, whether the function is concave up or down.

Applications of Derivatives

  • Derivatives have various applications in real-world situations, such as physics, engineering, economics, etc.
  • In many problems involving rates of change or maximization/minimization problems, differentiation is involved.
  • The derivative at a point can be used to find the tangent to the curve at that point and the normal, which is perpendicular to the tangent.