To this end we look at the change in position as the change in time approaches 0. You may have noticed that the two operations we've discussed — computing the slope of the tangent to the graph of a function and computing the instantaneous rate of change of the function — involved exactly the same limit.
This function is known as the derivative of the original function. Since lots of different sorts of people use derivatives, there are lots of different mathematical notations for them. Here are some:. The first notation has the advantage that it makes clear that the derivative is a function.
This is consistent with the definition of the derivative as the slope of a function. We can do it "the hard and imprecise way", without using differentiation, as follows, using a calculator and using small differences below and above the given point:.
We were lucky this time; the approximation we got above turned out to be exactly right. But this won't always be so, and, anyway, this way we didn't need a calculator.
This sort of point of non-differentiability is called a cusp. Functions may also not be differentiable because they go to infinity at a point, or oscillate infinitely frequently.
Wikipedia has related information at Notation for differentiation. The derivative notation is special and unique in mathematics. Either way is a good way of thinking, although you should remember that the precise definition is the one we gave above. The process of differentiation is tedious for complicated functions. Therefore, rules for differentiating general functions have been developed, and can be proved with a little effort. Once sufficient rules have been proved, it will be fairly easy to differentiate a wide variety of functions.
Some of the simplest rules involve the derivative of linear functions. Since we already know the rules for some very basic functions, we would like to be able to take the derivative of more complex functions by breaking them up into simpler functions.
Two tools that let us do this are the constant multiple rule and the addition rule. The details are left as an exercise. The fact that both of these rules work is extremely significant mathematically because it means that differentiation is linear. You can take an equation, break it up into terms, figure out the derivative individually and build the answer back up, and nothing odd will happen. We now need only one more piece of information before we can take the derivatives of any polynomial.
This has been proved in an example in Derivatives of Exponential and Logarithm Functions where it can be best understood. Since polynomials are sums of monomials, using this rule and the addition rule lets you differentiate any polynomial.
A relatively simple proof for this can be derived from the binomial expansion theorem. With these rules in hand, you can now find the derivative of any polynomial you come across. Rather than write the general formula, let's go step by step through the process.
These are not the only differentiation rules. There are other, more advanced, differentiation rules, which will be described in a later chapter. From Wikibooks, open books for an open world. Can you see a pattern? Use the definition of the derivative to show this. Category : Book:Calculus. Introduction to Calculus , where there is a brief history of calculus. The Derivative , an introduction to differentiation, for those who have never heard of it.
Differentiation of Transcendental Functions , which shows how to find derivatives of sine, cosine, exponential and tangential functions. Integration , which is actually the opposite of differentiation. Differential Equations , which are a different type of integration problem, but still involve differentiation. Tangents and Normals which are important in physics eg forces on a car turning a corner. Newton's Method - for those tricky equations that you cannot solve using algebra. Curvilinear Motion , which shows how to find velocity and acceleration of a body moving in a curve.
Related Rates - where 2 variables are changing over time, and there is a relationship between the variables. Curve Sketching Using Differentiation , where we begin to learn how to model the behaviour of variables. More Curve Sketching Using Differentiation.
Applied Maximum and Minimum Problems , which is a vital application of differentiation. Radius of Curvature , which shows how a curve is almost part of a circle in a local region.
Name optional. Tangents and Normals 2. Curvilinear Motion 4. Related Rates 5. Curve Sketching Using Differentiation 6.
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