Differentiation Rules 4628Fd

1. **Stating the problem:** We want to understand differentiation and all the key rules used in IGCSE Further Pure Mathematics. 2. **What is differentiation?** Differentiation is the process of finding the derivative of a function, which measures how the function changes as its input changes. It tells us the rate of change or slope of the function at any point. 3. **Basic differentiation formula:** For a function $f(x) = x^n$, the derivative is given by $$\frac{d}{dx} x^n = n x^{n-1}$$ where $n$ is any real number. 4. **Important differentiation rules:** - **Constant rule:** The derivative of a constant $c$ is zero. $$\frac{d}{dx} c = 0$$ - **Sum rule:** The derivative of a sum is the sum of the derivatives. $$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$ - **Difference rule:** The derivative of a difference is the difference of the derivatives. $$\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$$ - **Product rule:** For two functions $u(x)$ and $v(x)$, $$\frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)$$ - **Quotient rule:** For two functions $u(x)$ and $v(x)$, $$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) u'(x) - u(x) v'(x)}{[v(x)]^2}$$ - **Chain rule:** For a composite function $f(g(x))$, $$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$$ 5. **Examples:** - Differentiate $f(x) = 3x^4$. Using the power rule: $$f'(x) = 3 \times 4 x^{4-1} = 12 x^3$$ - Differentiate $y = (2x + 1)^5$. Using the chain rule: Let $u = 2x + 1$, then $y = u^5$. $$\frac{dy}{dx} = 5 u^{4} \times \frac{du}{dx} = 5 (2x + 1)^4 \times 2 = 10 (2x + 1)^4$$ 6. **Summary:** Differentiation helps find rates of change. The power, constant, sum, difference, product, quotient, and chain rules are essential tools. Practice applying these rules to different functions to master differentiation.