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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.

4 Leonard Euler (1707–1783), one of the great mathematicians of the 18th century. 28 CHAPTER 1. SOME BACKGROUND MATERIAL Exponential Functions grow fast. 21 (Exponential Growth). It is not so apparent from the graph how fast the exponential function grows. , doubling the number of grains with each square. The chess board has 64 squares, and that commits him to 263 grains on the 64th square for a total of 264 − 1 = 18, 446, 744, 073, 709, 551, 615 grains. In mathematical notation, you say that he puts f (n) = 2n−1 grains on the n-th square of the chess board.

3. DERIVATIVES OF SOME BASIC FUNCTIONS 51 with absolute values. 14), and in the last one we use that | sin x0 | and | cos x0 | are ≤ 1. | sin(x0 + h) − [sin x0 + (cos x0 )h]| = | sin x0 cos h + sin h cos x0 − sin x0 − (cos x0 )h| = | sin x0 (cos h − 1) + cos x0 (sin h − h)| ≤ | sin x0 || cos h − 1| + | cos x0 || sin h − h| ≤ | sin x0 | ≤ h2 . 13) with our chosen A and d. In particular, we have shown that the sine function is differentiable and that its derivative is the cosine function. Let us look at the specific value x0 = π/4.

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