1. **Stating the problem:** We want to understand what a derivative is and see some examples. 2. **Definition:** The derivative of a function measures how the function's output changes as its input changes. It represents the slope of the tangent line to the function's graph at any point. 3. **Formula:** The derivative of a function $f(x)$ at a point $x$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This limit, if it exists, gives the instantaneous rate of change of $f$ at $x$. 4. **Important rules:** - The derivative of a constant is 0. - The derivative of $x^n$ is $nx^{n-1}$. - The derivative of a sum is the sum of derivatives. 5. **Example 1:** Find the derivative of $f(x) = x^2$. Using the power rule: $$f'(x) = 2x^{2-1} = 2x$$ This means the slope of the curve $y=x^2$ at any point $x$ is $2x$. 6. **Example 2:** Find the derivative of $f(x) = 3x^3 + 5x$. Using the sum and power rules: $$f'(x) = 3 \cdot 3x^{3-1} + 5 \cdot 1x^{1-1} = 9x^2 + 5$$ 7. **Interpretation:** The derivative tells us how fast the function is changing at any point. For $f(x) = x^2$, at $x=2$, the slope is $2 \times 2 = 4$, meaning the function is increasing at a rate of 4 units per unit increase in $x$. **Final answer:** The derivative is a fundamental concept in calculus that gives the rate of change or slope of a function at any point. Examples include $\frac{d}{dx} x^2 = 2x$ and $\frac{d}{dx} (3x^3 + 5x) = 9x^2 + 5$.