Differentiation

Introduction to Differentiation:

Definition: Differentiation is a fundamental concept in calculus that involves finding the rate at which a quantity changes concerning another variable. It determines how a function changes as its input changes.

Derivative Notation: The derivative of a function f(x) with respect to x is denoted by f(x) or dfdx.

Basic Differentiation Rules:

  1. Power Rule:

    • For a function f(x)=xn, the derivative is f(x)=nxn1.
  2. Constant Multiple Rule:

    • If c is a constant and f(x) is differentiable, then ddx[cf(x)]=cddx[f(x)].
  3. Sum and Difference Rule:

    • The derivative of the sum/difference of two functions is the sum/difference of their derivatives: ddx[f(x)±g(x)]=ddx[f(x)]±ddx[g(x)].
  4. Product Rule:

    • For two functions u(x) and v(x), the derivative of their product is: ddx[u(x)v(x)]=u(x)v(x)+u(x)v(x).
  5. Quotient Rule:

    • For u(x) and v(x) where v(x)0, the derivative of their quotient is: ddx[u(x)v(x)]=v(x)u(x)u(x)v(x)[v(x)]2.

Special Derivatives:

  • Trigonometric Functions:

    • ddx[sin(x)]=cos(x)
    • ddx[cos(x)]=sin(x)
    • ddx[tan(x)]=sec2(x)
  • Exponential and Logarithmic Functions:

    • ddx[ex]=ex
    • ddx[ln(x)]=1x

Chain Rule:

Chain Rule:

  • If y=f(u) and u=g(x), then the chain rule states that dydx=dydududx.

Implicit Differentiation:

Implicit Differentiation:

  • When an equation is given implicitly in terms of x and y, you can differentiate both sides of the equation with respect to x and solve for dydx.

Higher Order Derivatives:

Higher Order Derivatives:

  • The second derivative of a function f(x) represents the rate of change of its first derivative with respect to x. It's denoted as f(x) or d2fdx2.

Applications of Differentiation:

  • Optimization: Finding maximum or minimum values of functions.
  • Related Rates: Analyzing the rates at which quantities change concerning each other.
  • Curve Sketching: Determining behavior and characteristics of a function using derivatives.