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An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
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 \(a\).
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) \(f(x)=c\)
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)
Find equations of tangents to graphs of functions.
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.
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.
Solve practical problems concerning optimisation and rate of change, including calculus of motion.