Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Define differentiation for transcendental functions.
• Apply rules to find derivatives of exponential functions (like e to the power of x).
• Apply rules to find derivatives of logarithmic functions (like the natural log of x).
• Solve problems involving these functions.

Lesson summary

In our previous lessons, we learned how to find the rate of change of algebraic functions like polynomials. However, many real-world phenomena in Ghana and around the world don't follow simple polynomial patterns. Think about how the money in a savings account grows with compound interest, how Ghana's population increases over time, or even how hot kenkey cools down. These processes are often described by special functions called transcendental functions, specifically the exponential function (`e^x`) and the natural logarithmic function (`ln x`). Today, we will learn the techniques to differentiate these powerful functions.

Lesson notes

Part 1: What are Transcendental Functions? A transcendental function is a function that is not algebraic. This means it cannot be expressed as a finite sequence of algebraic operations (addition, subtraction, multiplication, division, raising to a power, and root extraction). The main types we will study are: Exponential functions: `f(x) = e^x`, `f(x) = 10^x` Logarithmic functions: `f(x) = ln(x)`, `f(x) = log(x)` Trigonometric functions: `f(x) = sin(x)`, `f(x) = cos(x)` (covered in a later topic)

Our focus today is on the natural exponential function (`e^x`) and the natural logarithmic function (`ln x`). The number `e` (Euler's number) is a special mathematical constant, approximately equal to 2.71828. It is special because the function `f(x) = e^x` has the unique property that its derivative is itself. Part 2: Derivative of the Natural Exponential Function (e^x)

Rule 1: The Basic Rule The derivative of `e^x` is simply `e^x`. > If `y = e^x`, then `dy/dx = e^x`.

Rule 2: The Chain Rule Application More often, the power of `e` is not just `x`, but another function of `x`, let's call it `u`. > If `y = e^u` where `u` is a function of `x`, then `dy/dx = (du/dx) * e^u`. > In simpler terms: Differentiate the power, and multiply it by the original function.

Evaluation guide

• Identify and apply techniques of differentiation to solve problems involving
• transcendental functions.
• Talk for Learning, Think -pair- share, Experiential Learning, and Group
• Work/Collaborative Learning.