1. Definition of a Derivative: Understanding the derivative as the limit of the average rate of change; the fundamental concept of differentiation.

2. Rules of Differentiation: Basic rules like the power rule, product rule, and quotient rule; essential for computing derivatives efficiently.

3. Chain Rule: Differentiating composite functions; crucial for handling nested functions.

4. Implicit Differentiation: Differentiating functions defined implicitly; useful for dealing with equations where y is not isolated.
5. Higher-Order Derivatives: Derivatives of derivatives (second, third, etc.); important for understanding concavity and acceleration.
6. Derivatives of Trigonometric Functions: Differentiating sine, cosine, and other trigonometric functions; widely used in oscillatory and wave phenomena.
7. Derivatives of Exponential and Logarithmic Functions: Understanding the behaviour of growth and decay processes.
8. Applications of Derivatives: Using derivatives to solve problems involving rates of change, optimisation, and motion.
9. Mean Value Theorem: Relates the derivative of a function to its average rate of change over an interval; fundamental for proving properties of functions.
10. [[L’Hôpital’s Rule]]: A method for evaluating limits involving indeterminate forms; leverages derivatives for solving complex limit problems.
11. Taylor Series and Polynomial Approximations: Using derivatives to approximate functions locally with polynomials; important for numerical analysis.
12. Curve Sketching: Using derivatives to understand the shape and behaviour of graphs of functions, including critical points and inflection points.

Finding the tangent line of a function at a point $a$

straight line is $𝑦=𝑚𝑥+𝑐$ $m$ can be found by differentiating $f(x)$ and passing in the value of x $m=f’(a)$

the tangent lies at the point $(a, f(a))$ so we find $f(a)$ and solve for $c$

$$f(a) = ma + c$$

Problems: Find the tangent line to $𝑓(𝑥)=\frac{16}{𝑥}−4\sqrt𝑥$ at $𝑥=4$

$$f’(x) = \frac{0x-161}{x^2} - \frac{4}{2}x^{\frac{-1}{2}} = \frac{-16}{x^2} - \frac{2}{\sqrt{x}}$$ $$f’(4)=\frac{-16}{16}-\frac{2}{2}=-2 = m$$

$$f(4) = \frac{16}{4} - 4\sqrt{4} = 4-8 = -4$$

$$-4 = -2*4 + c$$ $$c=4, \text{tangent} = -2x + 4$$