Problem 1: Consider some arbitrary function f(x), whose value and whose derivative values of every order are already known at some point ro. We can approximate the function value f(x) with f(xo)+f'(xo)(x–x0) for points a that are very close to ro- (a) Justify this "linear" approximation using the graphical interpretation of the derivative. Because this is an approximation, there will be a discrepancy between the actual f(x) and our approximation – call this g(x), and write down an expression for f(x) including g(x). (The approximation is called linear due to how the (æ – xo) term is first-order.) You may use the graph of f (x) = e* , with your own choice of ro, as an illustrative example.

The curve y = ax + bx +c passes through the point (1,8) and is tangent to the line y = 5x at the origin. Find a, b, and c. (Hint: You can find these numbers by doing a few things. You can plug (1,6) and the origin into y. You can find the derivative of y=f(x); what is the derivative of f(x) at the origin?

please help me to find the derivative of this question

Suppose we are asked to compute the 100th derivative of x 3 . How can we obtain the answer in a fast and easy way? What is the final answer to this problem?