THE NATURAL EXPONENTIAL FUNCTION

  1. DEFINITION AND PROPERTIES

  • The natural exponential function f(x) = e^x is the inverse of the natural logarithmic function f(x) = ln x.

  • Inverse Relationship: y = e^x if and only if x = ln y.

  • Cancellation Equations: ln(e^x) = x and e^(ln x) = x.

  • Graph Characteristics:

    • Domain: (-infinity, infinity)

    • Range: (0, infinity)

    • The graph is continuous, always increasing, and concave upward on its entire domain.

  • Limits:

    • As x approaches negative infinity, e^x approaches 0 (Horizontal Asymptote).

    • As x approaches positive infinity, e^x approaches infinity.

  1. DIFFERENTIATION

  • Base Rule: The derivative of e^x is e^x. It is the only non-zero function that is its own derivative.

  • Chain Rule: If the exponent is a function (u), you must apply the chain rule: d/dx [e^u] = e^u * (du/dx).

  • Example: The derivative of e^(5x) is 5e^(5x).

  1. INTEGRATION

  • Base Rule: The integral of e^x dx = e^x + C.

  • Linear Shortcut: For a constant k, the integral of e^(kx) dx = (1/k)e^(kx) + C.

  • U-Substitution: For complex exponents, set u equal to the exponent.

  • Example from Video: In the integral of x^3 * e^(x^4), let u = x^4. The du = 4x^3 dx allows the x^3 term to be cancelled out.

  • Algebraic Tip: If the integrand is a fraction with a single term in the denominator (like e^x), split the fraction into separate terms before integrating.