1. Let's start with differentiation, which is the process of finding the derivative of a function. The derivative represents the rate of change or slope of the function at any point. 2. The basic formula for differentiation is: $$\frac{d}{dx} x^n = nx^{n-1}$$ where $n$ is any real number. 3. Important rules include the sum rule: $$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$ and the product rule: $$\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$. 4. For example, differentiate $f(x) = 3x^4 + 5x^2$: - Apply the power rule to each term: $$\frac{d}{dx} 3x^4 = 3 \times 4x^{3} = 12x^{3}$$ $$\frac{d}{dx} 5x^2 = 5 \times 2x^{1} = 10x$$ - So, $f'(x) = 12x^{3} + 10x$. 5. Now, integration is the reverse process of differentiation. It finds the original function given its derivative. 6. The basic formula for integration is: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1$ and $C$ is the constant of integration. 7. For example, integrate $g(x) = 6x^{2} + 4x$: - Integrate each term: $$\int 6x^{2} dx = 6 \times \frac{x^{3}}{3} = 2x^{3}$$ $$\int 4x dx = 4 \times \frac{x^{2}}{2} = 2x^{2}$$ - So, the integral is $G(x) = 2x^{3} + 2x^{2} + C$. 8. Remember, differentiation and integration are fundamental tools in calculus used to analyze functions and their behavior.