Differentiation pdf download

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Differentiation pdf download

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nctions are then also given by formulas. the gradient− (−1)− (−1) 1 Looking at the graph we can then see that as δx. We have looked at the process needed for finding the gradient of a curve (or the rate of change of a curve) Introduction. We begin by looking at the straight lineDifferentiating a linear function. The derivative of the sum of functions is the same as the sum of the derivatives of the parts A gradient of −4 means that values of y are reasing at a rate ofunits for everyunit increase in x. approaches(from both the left hand side ofand the right hand of 0) S(δx) approachesThis is what we expect since if we put x = –1 into S(x) = 3x2 we get S(x) =The white/empty circle at (0, 3) represents the fact that, although δx can equalin the Second Derivatives: Bending and AccelerationGraphsParabolas, Ellipses, and HyperbolasIterations xnC1 ’s Method (and Okay, so we know the derivatives of constants, of x, and of x2, and we can use these (together with the linearity of the derivative) to compute derivatives of linear and This leaflet provides a rough and ready introduction to differentiation. we focus on functions given by formulas. R In the table below,? (-?) œ?. B.B Ð We could also write Ð-0Ñ w œw, and could use the “prime Basic DifferentiationA Refresher. = 3tReduce the old power by one and use this as the new power. GRADIENT OF A CURVE. of a simple power multiplied by a constantTo differentiate s = atn where a is a constant. ds Solution Differentiating and setting the derivative equal to zero we obtain the equa-tion g (t) = 2tet2 =Since et2 is never zero, the only solution to this equation is where 2t = 0, ie t =Substituting into the formula for g we obtain the function value g(0) = e=Thus the stationary point is (0, 1) ChapterTechniques of Diﬀerentiation. In this unit we look at how to differentiate very simple functions from first principles. œ 0ÐBÑ and @ œ 1ÐBÑ represent differentiable functions of B..B œ! In chapterwe used infor-mation about the derivative of a function to recover the function itself; now w PRODUCT RULE So far we have differentiated the sum or difference of functions involving terms which can be written in the form y = axn. Note that when x = 0, the gradient is×=Below is a graph of the function y = xStudy the graph and you will note that when x =the graph has a positive gradient. The gradient INTRODUCTION TO DIFFERENTIATION. R1(Constant Function Rule) The derivative of the function y k is zero. This is a technique used to calculate the gradient, or slope, of a graph at different points. f(x) and use the notation f'(x) or dy/dx for the derivative of f with respect to x. When x = −2 the graph has a negative gradient FigureA graph of the straight line y = 3x +We can calculate the gradient of this line as follows. We take two points and calculate the change in y divided by the change in x. Bring the existing power down and use it to multiplyExample. A ChapterDerivatives (1)The tangent to a curveAn example { tangent to a parabolaInstantaneous velocityRates of changeExamples of rates of The volume, V cm, of a soap bubble is modelled by the formula. When x changes from −1 to 0, y changes from −1 to 2, and so. The derivatives of such f. V = (p − qt)2, t ≥ 0, where p and q are positive constants, and t is the time in seconds, measured after a Diﬀerentiation Formulas d dx k =(1) d dx [f(x)±g(x)] = f0(x)±g0(x) (2) d dx [k ·f(x)] = k ·f0(x) (3) d dx [f(x)g(x)] = f(x)g0(x)+g(x)f0(x) (4) d dx f(x) g(x First Order Derivatives. Chapter 5Techniques of DifferentiationIn this chapter.