1. Let's start by stating the problem: We want to understand the concept of differentiation in IBDP Maths AI SL calculus. 2. Differentiation is the process of finding the derivative of a function, which represents the rate of change or slope of the function at any point. 3. The basic formula for the derivative of a function $f(x)$ is given by: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This formula calculates the instantaneous rate of change of $f(x)$ as $h$ approaches zero. 4. Important rules to remember: - Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. - Constant Rule: The derivative of a constant is zero. - Sum Rule: The derivative of a sum is the sum of the derivatives. - Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$. - Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $$f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$$ - Chain Rule: If $f(x) = g(h(x))$, then $$f'(x) = g'(h(x)) \cdot h'(x)$$ 5. Let's do an example: Differentiate $f(x) = 3x^4 - 5x^2 + 7$. 6. Using the power rule and constant rule: $$f'(x) = 3 \cdot 4x^{4-1} - 5 \cdot 2x^{2-1} + 0 = 12x^3 - 10x$$ 7. This means the derivative of $f(x)$ is $f'(x) = 12x^3 - 10x$, which gives the slope of the curve at any point $x$. 8. Differentiation helps us analyze graphs, find maxima and minima, and solve many real-world problems involving rates of change.