1. Let's start by stating the problem: We want to understand differentiation, which is a way to find how a function changes as its input changes. 2. 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 means we look at the average rate of change of the function over a very small interval $h$, and then see what happens as $h$ gets closer and closer to zero. 3. Important rules to remember: - The derivative of a constant is 0. - The derivative of $x^n$ (where $n$ is a number) is $nx^{n-1}$. - The derivative of a sum is the sum of the derivatives. 4. Let's do an example: Find the derivative of $f(x) = x^2$. 5. Using the power rule, the derivative is: $$f'(x) = 2x^{2-1} = 2x$$ 6. This means the slope of the function $x^2$ at any point $x$ is $2x$. 7. Differentiation helps us find slopes of curves, rates of change, and is fundamental in calculus. Keep practicing with different functions to get comfortable!