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: 9
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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In our previous lessons, we learned how to find the derivative of simple functions, like polynomials, using the power rule. This is a powerful tool for finding rates of change, like the gradient of a curve at a point. However, in the real world, functions are often more complex. They can be combinations of other functions being multiplied together or divided by each other. For instance, a company's revenue is the product of the number of items sold and the price per item. An engineer might analyse the efficiency of a system which is a ratio of output to input. To analyse the rate of change in these situations, we need more advanced rules.
A. Recap: The Power Rule and Basic Derivatives
Before we introduce new rules, let's remember what we already know. The power rule states that if `f(x) = axⁿ`, then its derivative `f'(x) = n * axⁿ⁻¹`. Example 1: If `y = 4x³`, then `dy/dx = 3 * 4x³⁻¹ = 12x²`. Example 2: If `f(x) = √x = x¹/²`, then `f'(x) = (1/2)x¹/²⁻¹ = (1/2)x⁻¹/² = 1 / (2√x)`.
We also know the derivatives of basic trigonometric functions: `d/dx (sin x) = cos x` `d/dx (cos x) = -sin x`
Important Misconception: If we want to differentiate `f(x) = x² * sin(x)`, we CANNOT just differentiate each part and multiply them. That is, `f'(x) ≠ (2x) * (cos x)`. This is incorrect! We need a specific rule for products. B. The Product Rule