Author: haroonkhan

The derivative tells us something about the rate at which a function f increases or decreases at some point x. This rate is the slope of the tangent line to the graph of f at x. So, the derivative tells us something about the “linear” behavior of a function. However, this does not tell us anything about its curvature. To visualize…

The derivative is the slope of the tangent line to the graph of a function. Hence, the sign of the derivative at a point tells us whether the function is increasing or decreasing there and how rapidly. We can use this to figure out essential features of a function and to sketch its graph. Example…

The rules of previous section do not help us in finding derivatives of functions like the square root or, given the derivative of logarithm, in finding the derivative of the exponential. We need a rule to deal with the derivative of an inverse function. THEOREM 2.9 (Derivative of an inverse function) Let x = g(y) be the…

Given two functions g and h, we may build a new function by composition, namely, g o h. It would be nice to have a way of finding the derivative of the composite function by decomposing the task and exploiting knowledge about the derivatives of g and h. THEOREM 2.8 (Chain rule) Given functions g and h, we obtain the derivative of their composition as…

Given two functions f and g, there are a few easy ways to build other functions by ordinary arithmetic operations such as sum, multiplication, and division. If we are able to find the derivative of f and g, the following theorem shows how to find the derivative of functions defined by the mechanisms above. THEOREM 2.7 Let f and g be functions…

The direct application of the definition to find the derivative of a function is typically a rather difficult and cumbersome procedure, possibly requiring some intuition. Example 2.19 (Derivative of logarithm and exponential function) One of the most useful results concerning derivatives is that the derivative of the exponential is just the exponential itself: As a first…

If the derivative of function f at point Xo exists, then we say that the function is differentiable at point x0; if this holds for all points on an interval or domain, the function is differentiable on that interval or domain. If the derivative f′(x) exists at all points x on an interval and the derivative is a continuous function, we say that…

Consider a point x0 and the increment ratio of function f at that point: Fig. 2.17 The derivative is the limit of an increment ratio. For a nonlinear function, keeping x0 fixed, this ratio is a function of h. Now consider smaller and smaller steps h, as illustrated in Fig. 2.17. If we let h → 0, we get the “tangent” line to the graph of f at point x0.…

We have seen that a linear (affine) function f(x) = mx + q has a well-defined slope. Whatever value of the independent variable we consider, the slope of the function is always the same. If we are at point x and we move to point x + h, by any displacement h, the increment ratio14 is: Fig. 2.16 A nonlinear function does not have constant increment ratios.…

The logarithm arises as the inverse of an exponential function. To further motivate this, let us consider again continuous compounding of interest rates. As we have pointed out, continuous compounding leads to an exponential function that streamlines financial calculations considerably. However, in practice, interest rates are not quoted like this. Typically, interest rates are quoted…