Grade 12 Calculus and Vectors
Chapter 6: Derivatives of Exponential Functions

The general form of the exponential function is f(x)=b^x, and a special form of this function is of the form f(x)=e^x. Both are used mostly to model rapid change in one element with respect to another. For example, exponential functions are used to model the rapid growth and decay of real life situations and scientific studies. In Calculus and Vectors, these two functions are used to represent growth and decay of bacteria culture, economies, human population, environmental sustainability, carbon decay in decomposing bodies, half-life of certain radiating elements, and amortization^† calculations, to name a few.

As in polynomials, this function can also be transformed in different ways to depict unique situations, so students should review the idea of exponential functions of the form f(x)=a*M^(b(x+c)) + d. Where,

a = vertical stretch/compression factor (a>1 = stretch and 0<a<1 = compression)
b = horizontal stretch/compression factor (0<b<1 = stretch and a>1= compression)
c = horizontal shift factor (+c = shift left and –c = shift right)
d = vertical shift factor (+d = shift up and –d = shift down)

Moreover, the inverse of exponential functions f(x)=b^x and f(x)=e^x are f(x)=log_bx and f(x)=ln x ,^‡ respectively. Most often, the combinations are used together to simplify or counteract each other in calculations. In addition, f(x)= ln x is a special form of the logarithmic function f(x)=log_bx, which is f(x)=log_ex, where base "e" is an irrational number called the "natural number." It also has an approximate numerical value of 2.72.


6.1 Derivative of Exponential Functions f(x)=e^x



6.2 Derivative of General Exponential Functions f(x)=b^x



6.3 Optimization Problems - Exponential Functions