Definition of Derivative A tangent line is the straight line that touches a curve at just one point. Tangent lines arecentral to calculus and have been since the 17th century. Today, we'll examine the problem of determining tangent lines for functions. The two fundamental questions addressed in calculus are: (1) What is the area under a curve,and (2) what is the slope of a line? In Algebra I, we learned that the slope of a function equals the change in y divided by thechange in x. The slope of f(x) = x3 − 4x at x = 1 is not equal to this expression because we areat a point on our function where y changes sign. If we take the reciprocal of both sides of the identity above and simplify, we get 1/f of 1 - 1/f of1 over 1 - 1. This simplifies to 0/0, which is indeterminate form. We know from limits that 0/0 isan indeterminate form because we don't know what to make of 0/0. We might get a numericalvalue like 0 or we might get it does not exist. We can't tell without further investigation. To solve this problem, we are going to find successive average rates of change between pointsthat are around 1. We will go from 0 to 1-- 0.5 to 1, 0.75 to 1, until we close in on the x value ofexactly 1. So we are finding the slope between two points that are so close together they mightas well be the same point. All right, let's execute. As we have seen, we find the slope of a straight line at given intervals and close in on theslope so that the difference between the two points is so small that they might as well beequal. They can be even 1 and 1. As promised, the values have been automatically populated. There is no need to go through allthat. The process is simple math—crunching numbers. I want to really get into the calculushere, so let's take a look at what's happened. We started with an interval of 0 to 1 and got -3.As we get closer and closer to an interval length of 0 (which we are doing here), we get closerand closer to this x value being 0. The slope of the tangent line is equal to the negative reciprocal of the slope of the secant line.If we know that the slope of the secant line is 1, then we can find the slope of the tangent lineby solving for r in this equation: r=1/m=-1. This is a critical piece of information, because itallows us to determine that a point on an inverse function is always (-1) times as far from anx-intercept as it is from an y-intercept. In other words, if a point on an inverse function lies 1unit from an x-intercept, then it lies (-1) units from a y-intercept; conversely, if a point on aninverse function lies (-1) units from a y-intercept, then it lies 1 unit from an x-intercept. Next, I am going to graph a quadratic function and call the point where the graph intersects thex-axis x. This is point x, which means this is the coordinate x( f(x)) of x. And then this pointright here is so close to it that they are 0.9 and 1. They are very, very close together. So closethat I'm not even going to put a value on the difference between these two values. I'm justgoing to call it some horizontal distance h. That means this coordinate is x plus h and f(x) plush are the two coordinates for this graph. We can find the slope of the tangent line at any point on a curve by finding the derivative of thefunction at that point and dividing by the difference between the x-coordinates of that point andthe point on the curve at which we want to know the slope. In the previous example, we took a limit of the slope of the secant lines to zero as the numberof observations increased. In this example, we will take a limit of the slope of the tangent lineas it approaches zero.
What is happening is that, as this horizontal distance is becoming smaller and smaller,approaching 0, these two points are practically the same. Again, if you evaluate this limit whenh equals 0, you would get f of x minus f of x over 0/0. This means that you have to do somealgebra to clear up the 0/0. This turns out to be an essential formula in calculus. A derivative is the instantaneous rate of change at a point on a curve. The slope of the tangentline at a given point is its derivative. There are two formulas for finding the slope of a curve at a specific point. The formula you usedepends on whether you want to find the derivative of a function or the derivative of an inversefunction. In calculus, there are two ways to express the derivative of a function. The derivative isrepresented by the symbol d and the variable of the function's independent variable raised tothe first power. The function's dependent variable is represented by x. This way ofrepresenting derivatives is more like an operation, as in take the derivative of this function,much like you might see multiply these two functions together or divide these two functions.Whereas, this notation--f prime of x--is more of a name. It is the name of the derivative of agraph or function f.