Check list:

1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Use limits to define the derivative of a function \(f\) at any \(x\): \(f'( x ) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\) Generalise to find the derivative of \(f\) at any point \(x\) in the domain of \(f\), i.e. define the derivative function \(f'( x )\) of the function \(f(x)\). Understand intuitively that \(f'( a)\) is the gradient of the tangent to the graphs of \(f\) at the point with \(x\)-coordinate \(a\).
3. Using definition (first principle), find the derivative, \(f'( x) \) for \(a\), \(b\) and \(c\) constants:
   (a)  \(f(x)=ax^{2}+bx+c\);
   (b)  \(f(x)=ax^3\)
   (c)  \(f\left( x \right) = \frac{a}{x}\)
   (d)  \(f(x)=c\)
4. Use the formula \(\frac{d}{{dx}}\left( {a{x^n}} \right) = an{x^{n - 1}}\) (for any real number \(n\)) together with the rules:
   (a)  \(\frac{d}{{dx}}\left[ {f(x) \pm g(x)} \right] = \frac{d}{{dx}}\left[ {f(x)} \right] \pm \frac{d}{{dx}}\left[ {g(x)} \right]\)
   (b)  \(\frac{d}{{dx}}\left[ {kf(x)} \right] = k\frac{d}{{dx}}\left[ {f(x)} \right]\) (\(k\) a constant)
5. Find equations of tangents to graphs of functions.
6. Introduce the second derivative \(f''(x) = \frac{d}{{dx}}\left( {f'(x)} \right)\) of \(f(x)\)and how it determines the concavity of a function.
7. Sketch graphs of cubic polynomial functions using differentiation to determine the co-ordinate of stationary points, and points of inflection (where concavity changes). Also, determine the \(x\)-intercepts of the graph using the factor theorem and other techniques.
8. Solve practical problems concerning optimisation and rate of change, including calculus of motion.

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Mind Action Series: Mathematics 12 Textbook & Workbook (2013):
Exercise 1 Page 163
Exercise 2 Page 166
Exercise 3 Page 171
Exercise 4 Page 177
Exercise 5 Page 181
Exercise 6 Page 188
Exercise 7 Page 190
Exercise 8 Page 193
Exercise 9 Page 197
Exercise 10 Page 202
Exercise 11 Page 205
Exercise 12 Page 207
Exercise 13 Page 210
Exercise 14 Page 211
Revision Exercise Page 212
Some Challenges Page 215

Relevant sections in Everything Maths:

• Limits
• Differentiation from first principles
• Rules for differentiation
• Equation of a tangent to a curve
• Second derivative
• Sketching graphs
• Applications of differential calculus
• Summary
• End of chapter exercises

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