## 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**. 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

^{x}^{†}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**. Where,

^{(b(x+c))}+ da = 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**and

^{x}**f(x)=e**are

^{x}**f(x)=log**and

_{b}x**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

_{b}x, which is

**f(x)=log**, where base "

_{e}x**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**

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

6.3 Optimization Problems - Exponential Functions

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

^{x}6.3 Optimization Problems - Exponential Functions

## Grade 12 Calculus and Vectors

Chapter 1: Introduction to* Calculus *and Vectors

Chapter 2: Derivatives

Chapter 3: Applications of Derivatives

Chapter 4: Curve Sketching

Chapter 5: Derivatives of Trigonometric Functions

Chapter 6: Derivatives of Exponential Functions

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 ^{.}

Chapter 7: Introduction to * Vectors *

Chapter 8: Vector Operations and Applications

Chapter 9: Equations of Lines

Chapter 10: Equations of Planes

Chapter 11: Points, Lines and Planes

Videos for each topic contain a playlist, allowing you to choose the instructor and learning style to your liking. You can choose different videos in a topic by placing the mouse over the video and selecting a different video from the playlist. Videos of solutions to multiple example problems are also available to illustrate to students how each tools and techniques should be used.

Please refer to high school math prerequisite tree to see the recommended courses and materials one should be familiar with prior to taking up Calculus and Vectors.

^{†}Amortization calculations use modified forms of exponential functions, to depict a less rapid growth and/or decay of asset values or principals on payments and etc.

^{‡} f(x) = log_{e}x = ln x; hence log_{e} = ln

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