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IB Mathematics AI SL The derivative of functions MAI Study Notes - New Syllabus

IB Mathematics AI SL The derivative of functions MAI Study Notes

LEARNING OBJECTIVE

  • The derivative of functions

Key Concepts: 

  • Differentiating Powers
  • Differentiating Polynomials

MAI HL and SL Notes – All topics

DIFFERENTIATING POWERS

Differentiating Powers

When differentiating a term of the form \( f(x) = ax^n \), where:

  • \( a \) is a constant coefficient
  • \( n \in \mathbb{R} \) (but most commonly an integer)

we apply the Power Rule:

Formula: \( \frac{d}{dx}(ax^n) = anx^{n-1} \)

This means you multiply the exponent \( n \) with the coefficient \( a \), and then subtract 1 from the exponent.

 Trick to Remember: Think of “bringing the power down” and reducing it by 1.

 Example: \( f(x) = 5x^3 \) becomes \( f'(x) = 15x^2 \)

 Why It Works: The power rule comes from the limit definition of the derivative, but is used because it simplifies calculations for polynomials.

Example

Differentiate: \( f(x) = 7x^5 \)

▶️Answer/Explanation

Solution:

  • Apply the Power Rule: \( f'(x) = 7 \cdot 5x^{4} \)
  • Result: \( f'(x) = 35x^4 \)

DIFFERENTIATING POLYNOMIALS

Differentiating Polynomials

A polynomial is an expression made by summing terms of the form \( ax^n \). For example:

\( f(x) = 4x^3 – 2x^2 + x – 6 \)

To differentiate a polynomial, apply the power rule to each term individually:

Formula: \( \frac{d}{dx}(ax^n + bx^m + \ldots) = anx^{n-1} + bmx^{m-1} + \ldots \)

Example

Differentiate: \( f(x) = 2x^4 – 3x^3 + 5x – 9 \)

  • Differentiate each term using the power rule:
  • \( f'(x) = 2 \cdot 4x^3 – 3 \cdot 3x^2 + 5 \cdot 1x^0 + 0 \)
  • Simplify: \( f'(x) = 8x^3 – 9x^2 + 5 \)

Example

Differentiate: \( f(x) = 2x^4 – 3x^3 + 5x – 9 \)

▶️Answer/Explanation

Solution:

  • Differentiate each term using the power rule:
  • \( f'(x) = 2 \cdot 4x^3 – 3 \cdot 3x^2 + 5 \cdot 1x^0 + 0 \)
  • Simplify: \( f'(x) = 8x^3 – 9x^2 + 5 \)

SPECIAL CASES IN DIFFERENTIATION

Special Cases in Differentiation

Differentiating Constants

The derivative of any constant is zero.

Rule: \( \frac{d}{dx}(c) = 0 \)

Example

Differentiate: \( f(x) = 10 \)

▶️Answer/Explanation

Solution:

  • Constant term → derivative is 0
  • \( f'(x) = 0 \)

Negative Integer Exponents

The power rule still works for negative powers of \( x \):

Rule: \( \frac{d}{dx}(x^{-n}) = -n x^{-n-1} \)

Example

Differentiate: \( f(x) = 4x^{-2} \)

▶️Answer/Explanation

Solution:
$-2\times 4 (x^{-2-1})$
$-8x^{-3}$

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