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# Differentiation

Syllabus Content

Understand and use the derivative of f (x) as the gradient of the tangent to the graph of y = f ( x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change, sketching the gradient function for a given curve, second derivatives, differentiation from first principles for small positive integer powers of x and for sin x and cos x. Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection

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## Furthermore

A local maximum is a point where a function has a higher value than at nearby points, making it a peak in the graph. Conversely, a local minimum is a point where the function has a lower value than its neighbours, forming a trough. To test for these points, we often use the first and second derivative tests. If $$f'(x)$$ changes sign from positive to negative as $$x$$ increases, $$f(x)$$ has a local maximum. If $$f'(x)$$ changes from negative to positive, $$f(x)$$ has a local minimum. The second derivative test states that if $$f''(x) > 0$$ at a point where $$f'(x) = 0$$, then $$f(x)$$ has a local minimum. If $$f''(x) < 0$$, it's a local maximum.

Optimisation involves finding the maximum or minimum values of a function in a given domain, which has numerous applications in real-world scenarios, such as maximising profit or minimising cost.

A point of inflexion is where a curve changes its curvature direction. If the curve changes from concave upwards to concave downwards (or vice versa), it's an inflexion point. At these points, $$f''(x) = 0$$ or is undefined. However, not all points where $$f''(x) = 0$$ are inflexion points. The gradient at an inflexion point can be zero (a horizontal tangent) or non-zero.

Examples:

1) Consider the function $$f(x) = x^3 - 3x^2$$.
The first derivative is $$f'(x) = 3x^2 - 6x$$ and the second derivative is $$f''(x) = 6x - 6$$.
Setting $$f'(x) = 0$$, we get $$x = 0$$ and $$x = 2$$. Using the second derivative test, we find that $$f(x)$$ has a local maximum at $$x = 0$$ and a local minimum at $$x = 2$$.

2) For the function $$g(x) = x^4 - 4x^3$$,
The first derivative is $$g'(x) = 4x^3 - 12x^2$$ and the second derivative is $$g''(x) = 12x^2 - 24x$$.
Setting $$g''(x) = 0$$, we get $$x = 0$$ and $$x = 2$$. Both of these are points of inflexion, but only $$x = 2$$ has a gradient of zero.

This video on the Basics of Differentiation is from Revision Village and is aimed at students taking the IB Maths Standard level course

This video on Optimization and Calculus Curves is from Revision Village and is aimed at students taking the IB Maths AA SL/HL course.

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