TMJC H2 Chapter 7 Differentiation Learning Package 2024

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Chapter 7 Differentiation TMJC 2024 Page 1 of 36 H2 Mathematics (9758) Chapter 7 Differentiation Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning Interpret the derivative d d y x as the gradient of a curve Use a graphing calculator to find the approximate value of a derivative at a given point Differentiate algebraic, trigonometric, logarithmic, exponential and inverse trigo functions Determine the nature of the stationary points (local maximum and minimum points and stationary point of inflexion) analytically, in simple cases, using the first derivative test or second derivative test Locate the maximum and minimum points using a graphing calculator Use chain rule, product rule and quotient rule Interpret graphically   f 0 x  (strictly increasing)   f 0 x  (stationary)   f 0 x  (strictly decreasing)   f 0 x  (concave upwards)   f 0 x  (concave downwards) Use a graphing calculator to investigate the relationship between the graphs of f( )y x and  f 'y x Differentiate functions defined implicitly Find higher derivatives Differentiate functions defined parametrically Relate the graph of  f 'y x to the graph of f( )y x

Chapter 7 Differentiation TMJC 2024 Page 2 of 36 §1 Differentiation In mathematics, differentiation is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integration, the study of the area under a curve. The primary objects of study in differentiation are the derivative of a function, related notions such as the differential, and their applications. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716), who provided independent and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration. Note: For d d d d yyx x   d dx : an operator to differentiate a function with respect to x  d d y x : a function which is the result of differentiating y with