1. The problem is to understand the concept of differentiation and how to find the derivative of a function. 2. Differentiation is the process of finding the rate at which a function is changing at any point. The derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{df}{dx}$. 3. The basic formula for the derivative of a function $f(x)$ is: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This formula represents the slope of the tangent line to the curve at point $x$. 4. Important rules for differentiation include: - Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$ - Constant Rule: $\frac{d}{dx} c = 0$ where $c$ is a constant - Sum Rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$ - Product Rule: $\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$ - Quotient Rule: $\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$ - Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$ 5. Example: Find the derivative of $f(x) = 3x^4 - 5x^2 + 6$ 6. Using the power rule and constant rule: $$f'(x) = 3 \cdot 4 x^{4-1} - 5 \cdot 2 x^{2-1} + 0 = 12x^3 - 10x$$ 7. So, the derivative of the function $f(x)$ is: $$f'(x) = 12x^3 - 10x$$ This derivative tells us the rate of change of the function at any value of $x$.