Lesson Notes By Weeks and Term v4 - SHS 2
PRINCIPLES OF CALCULUS
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PRINCIPLES OF CALCULUS
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Subject: Additional Mathematics
Class: SHS 2
Term: 2nd Term
Week: 8
Grade code: 2.3.1.LI.2
Strand code: 3
Sub-strand code: 1
Content standard code: 2.3.1.CS.1
Indicator code: 2.3.1.LI.2
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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This lesson introduces two powerful tools in differentiation: the Product Rule and the Quotient Rule. So far, we have learned to find the derivative (rate of change) of simple functions. But in the real world, quantities are often related by multiplication or division. For example, a farmer's total revenue is the price per yam multiplied by the number of yams sold. If both the price and the quantity sold are changing, how do we find the rate at which the revenue is changing? The Product Rule helps us answer this. Similarly, the efficiency of a machine might be its power output divided by its power input. The Quotient Rule helps us analyse how this efficiency changes.
Part 1: The Product Rule
(5 minutes) - Recap & Motivation
Let's quickly recall the Power Rule: `d/dx(ax^n) = anx^(n-1)`. Now, consider the function `f(x) = x⁵`. We know that `f'(x) = 5x⁴`. We can also write `f(x)` as a product, for example, `f(x) = x² * x³`. A common mistake is to think that the derivative of a product is the product of the derivatives. Let's test this: The derivative of `x²` is `2x`. The derivative of `x³` is `3x²`. Their product is `(2x)(3x²) = 6x³`.
Is `6x³` the same as our correct answer, `5x⁴`? No! This proves we need a special rule for differentiating products.