C10 Differentiation
对应大纲的内容
| MM6.1 | The derivative of $f(x)$ as the gradient of the tangent to the graph $y = f(x)$ at a point.
Differentiation from first principles is excluded. |
| MM6.2 | Differentiation of $x^{n}$ for rational $n$, and related sums and differences. This might require some simplification before differentiating.
For example, the ability to differentiate an expression such as $\frac{(3x+2)^{2}}{x^{\frac{1}{2}}}$. |
| MM7.3 | An understanding of the Fundamental Theorem of Calculus and its significance to integration. Simple examples of its use may be required in the forms:
|
学习目标
- Master pre-differentiation algebra by transforming complex rational functions and nested surds into linear combinations of $x^n$ to apply the power rule directly, bypassing the need for advanced quotient or product rules.
- Optimise gradients and higher-order rates by evaluating the second derivative $\left(f^{\prime \prime}(x)\right)$ not only to classify stationary points, but to precisely isolate the maximum or minimum possible gradient of a curve across a defined domain.
- Exploit the Fundamental Theorem of Calculus by transitioning seamlessly between integration and differentiation operators $\left(\frac{\mathrm d}{\,\mathrm d x} \int_a^x f(t) \mathrm{~d} t=f(x)\right)$ to extract integrands, synthesising this with the chain rule for composite upper limits.
- Execute parametric integral optimisation by differentiating definite integrals with respect to an external parameter (e.g., minimising $I(a)=$ $\int_0^1\left(x^2-a\right)^2 \mathrm{~d} x$ by differentiating with respect to $a$ ), treating $x$ strictly as a dummy variable.
建议时间
| Task | No. of Questions | Suggested Time |
|---|---|---|
| Examples | 7 | 60 min |
| Quiz | 5 | 10 min |
| Practice A | 26 | 40 min |
| Practice B | 16 | 40 min |
| Total | 54 | approx. 2-3 hrs |
Practice:
Practice:
