3.1, 3.2 Derivatives of powers and polynomials / Derivatives of exponentials and logs

Derivatives of powers and polynomials / Derivatives of exponentials and logs, 3.1, 3.2.

1) Main Points

Instead of finding the derivative by way of limits and points on graphs or tables, we can calculate the derivative by using an equation. For a linear function, we know that the rate of change is constant, and in y = mx + b form m = slope = f'(x). We also notice that for a particular x value, the derivative varies directly with the slope ie: y = 3f(x), slope = 3m, y = f(x)/2, slope = m/2. If a function is multiplied by a constant then the derivative is also multiplied by the same constant. When adding derivatives: f'(x) = {f(x) + g(x)} = f'(x) + f'(y). To find the derivative: {d/dx(x^n)= nx^n-1}. To find the slope of a tangent line at a certain point x on a function, you can find the derivative equation of the function and then substitute in the value of x. In 3.2 the book attempts to find the derivative of a power function. The find a value for f'(0), and use this as a multiplier. They also ask when (because both the derivative and the power function are increasing in the same direction) the derivative will be equal to the function. This happens with the function e^x. This helps us realize that : f'(a^x)= (ln a)a^x. For logarithmic functions: f'(lnx)= 1 / x.



2) Challenges

A challenge for me regarding derivatives in general is graphing them. It is important for me to remember that the derivative will have a very different shape than the function itself. And when the derivative is zero it means the graph is changing directions around a curve. Something I noticed about the nature of derivatives that may help me in the future is that the derivative is always one less power than the function (n-1). So if I am trying to find the derivative of a cubed function I know it will be a quadratic type line, etc. I am confused about why the book used f'(0) as a multiplier for the derivative of a power function. In this chapter I was introduced to a several ways of finding the derivative, so I will have to make sure I remember them and don't mix them up.

3) Reflections

Now we have expanded our knowledge of derivatives to encompass many of the types of functions we know today. This will be very useful for calculating the rates of change of those harder-to-model real life equations such as exponential and power functions, and logarithmic functions. Also inflation rates and equations that use natural logarithms.