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International Baccalaureate Mathematics

Calculus

Syllabus Content

Informal understanding of continuity and differentiability of a function at a point.
Understanding of limits (convergence and divergence).
Definition of derivative from first principles.
Higher derivatives.

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Furthermore

Official Guidance, clarification and syllabus links:

In examinations, students will not be asked to test for continuity and differentiability.

Link to: infinite geometric sequences

$$f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$

Use of this definition for polynomials only.

Familiarity with the notations \( \frac{d^ny}{dx^n}, \; f^{(n)}(x) \)

Link to: proof by mathematical induction

Formula Booklet:

Derivative of \(f(x)\) from first principles

$$y=f(x) \Rightarrow \frac{dy}{dx} = f'(x) = \lim_{h\to0} \left( \frac{f(x+h) - f(x)}{h} \right)$$

Differentiation from first principles is a method used to find the derivative of a function by computing the limit of the difference quotient as the change in the input variable approaches zero. In other words, it involves finding the gradient of a curve at a point by zooming in very closely and calculating the slope of the tangent line.

The key formula for differentiation from first principles is:

$$f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$

where \(f(x)\) is the function to be differentiated.

Here is an example of how to use differentiation from first principles to find the derivative of the function \(f(x) = x^2\):

$$\begin{aligned} f'(x) &= \lim_{h\to0} \frac{f(x+h) - f(x)}{h} \\ &= \lim_{h\to0} \frac{(x+h)^2 - x^2}{h} \\ &= \lim_{h\to0} \frac{x^2 + 2xh + h^2 - x^2}{h} \\ &= \lim_{h\to0} \frac{2xh + h^2}{h} \\ &= \lim_{h\to0} (2x + h) \\ &= 2x \end{aligned} $$

Therefore, the derivative of \(f(x) = x^2\) is \(f'(x) = 2x\).

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