By Dovermann K.H.

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7. 7, approximately. 7(x − 1) + e. CHAPTER 2. THE DERIVATIVE 38 Let us denote the slope of the tangent line to the graph of f at the point (x, f (x)) by f (x). Later on we will call f (x) the derivative of f at x and interpret f (x) as the slope of graph of f at (x, f (x)). 7. 12 on page 52. That means that the tangent line has the formula l(x) = e(x − 1) + e = ex. Our goal is to find a line which is close to the graph, near a given point. So let us check how close l(x) is to ex if x is close to 1.

2 in a less elegant but more practical way. ” Instead of asking for a line we ask for a number m, its slope, and use the line l(x) = f (x0 ) + m(x0 − x). 9. Let f be a function and x0 an interior point of its domain. 8) l(x) = f (x0 ) + m(x − x0 ). We denote its slope m by f (x0 ) and call it the derivative of f at x0 . We also say that f (x0 ) is the slope of the graph of f at x0 and the rate of change. To differentiate a function at a point means to find its derivative at this point. We provide one more reformulation which makes some calculations look more elegant.

With this choice of d it is assured that x + h ∈ (0, ∞) and that g(x + h) is defined. This is all we will need. We hope that you can recognize the steps in the following calculation. It is a challenge. 7 We use x instead of x0 . CHAPTER 2. THE DERIVATIVE 56 √ √ h x+h− x+ √ 2 x = = = = = ≤ = = ≤ √ √ h x− √ 2 x (x + h) − x h √ √ − √ 2 x x+h+ x 1 1 |h| √ √ − √ x+h+ x 2 x √ √ √ 2 x − ( x + h + x) |h| √ √ √ 2 x( x + h + x) √ √ x− x+h |h| √ √ √ 2 x( x + h + x) √ √ x− x+h |h| 2x x − (x + h) √ |h| √ 2x( x + x + h) 1 √ h2 √ 2x( x + x + h) 1 √ h2 2x x x+h− = Ah2 .