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Transcript of The Derivative
Fadhli Dzil Ikram (360)
Mohamad Rois (365)
Leonardus Andrew (370)
Azka Arrafi Aditama (375) The Derivative Group 15 - Calculus Definition of Derivative → The derivative of a function f is another function f' (f prime) whose value at any number c is: Provided that this limit exists and is not infinite EXAMPLES 2.2 If this limit does exist, we say that f is differentiable at c. Finding a derivative is called differentiation. The part of calculus associated with the derivative is called differential calculus. Equivalent Form For the Derivative c c+h (c+h, f(c+h)) (c, f(c) h f(c+h)-f(c) Equivalent Form For the Derivative c x (x, f(x)) (c, f(c) x-c f(x)-f(c) Differentiability Implies Continuity If a curve has a tangent line at a point, then that curve cannot take a jump or wiggle to badly at the point. The precise formulation of this fact is an important theorem. If f'(c) exists, then f is continuous at c. However, the converse of this theorem (every continuous line are differentiable) is false. f'(0) doesn't exists. f'(a) and f'(b) doesn't
exists. a b If the value of a variable x changes from x1 to x2, then x2 - x1, the change in x, is called an increment of x and is commonly denoted by delta x Increments (Delta) c c+h (c+h, f(c+h)) (c, f(c) h f(c+h)-f(c) LEIBNIZ NOTATION for the derivative As delta x moving approaches 0, Leibniz uses dy/dx in the equation instead limit of delta x approaching 0. This notation (dy/dx) is now a standard symbol for the derivative. This is the end of our presentation. Thanks for watching :) The Graph of the Derivative The derivative f'(x) gives the slope of the tangent line to the graph of y = f(x) at the value of x. Thus, when the tangent line is sloping up to the right, the derivative is positive, and when the tangent line is sloping down to the right, the derivative is negative. We can therefore get a rough picture of the derivative given just the graph of the function.