What is differentiation calculus




















Make sure that you can deal with fractional exponents. You will see a lot of them in this class. In all of the previous examples the exponents have been nice integers or fractions. They work exactly the same.

However, this problem is not terribly difficult it just looks that way initially. There is a general rule about derivatives in this class that you will need to get into the habit of using. When you see radicals you should always first convert the radical to a fractional exponent and then simplify exponents as much as possible. Following this rule will save you a lot of grief in the future.

There are some that we can do. It is still possible to do this derivative however. All that we need to do is convert the radical to fractional exponents as we should anyway and then multiply this through the parenthesis. So, as we saw in this example there are a few products and quotients that we can differentiate. If we can first do some simplification the functions will sometimes simplify into a form that can be differentiated using the properties and formulas in this section.

We know that the rate of change of a function is given by the functions derivative so all we need to do is it rewrite the function to deal with the second term and then take the derivative. Note that we rewrote the last term in the derivative back as a fraction. Again, notice that we eliminated the negative exponent in the derivative solely for the sake of the evaluation. Recall that if the velocity is positive the object is moving off to the right and if the velocity is negative then the object is moving to the left.

We need the derivative in order to get the velocity of the object. We can therefore write this definition of slope as:. An example may make this definition clearer. If you recall that the tangent of an angle is the ratio of the y-coordinate to the x-coordinate on the unit circle, you should be able to spot the equivalence here.

The graphs of most functions we are interested in are not straight lines although they can be , but rather curves. We cannot define the slope of a curve in the same way as we can for a line. In order for us to understand how to find the slope of a curve at a point, we will first have to cover the idea of tangency. Intuitively, a tangent is a line which just touches a curve at a point, such that the angle between them at that point is 0.

Consider the following four curves and lines:. A secant is a line drawn through two points on a curve. We can construct a definition of a tangent as the limit of a secant of the curve taken as the separation between the points tends to zero. Consider the diagram below.

The two points we draw our line through are:. Substituting in the points on the line,. This expression is called the difference quotient. The definition of the tangent line we gave was not rigorous, since we've only defined limits of numbers — or, more precisely, of functions that output numbers — not of lines.

But we can define the slope of the tangent line at a point rigorously, by taking the limit of the slopes of the secant lines from the last paragraph. Having done so, we can then define the tangent line as well. This formula gives the average velocity over a period of time, but suppose we want to define the instantaneous velocity. The derivative of a function is found by applying limits to the function as per the first principle of differentiation. For example, let us compute the derivative of sin x.

There are different rules followed in differentiating a function. The differentiation rules are power rule, chain rule, quotient rule, and the constant rule. We use the differentiation formulas to find the maximum or minimum values of a function, the velocity and acceleration of moving objects, and the tangent of a curve.

Learn Practice Download. Differentiation The process of finding derivatives of a function is called differentiation in calculus. What is Differentiation? Definition of Derivatives 3. What is Differentiation Formula? Rules of Differentiation 5. Differentiation of Special Functions 6. Higher-Order Differentiation 7. Partial Differentiation 8. Partial Differentiation The partial differential coefficient of f x,y with respect to x is the ordinary differential coefficient of f x,y when y is regarded as a constant.

Examples of Differentiation Example 1. Example 2. Great learning in high school using simple cues. Analyzing functions. Mean value theorem : Analyzing functions Extreme value theorem and critical points : Analyzing functions Intervals on which a function is increasing or decreasing : Analyzing functions Relative local extrema : Analyzing functions Absolute global extrema : Analyzing functions Concavity and inflection points intro : Analyzing functions. Analyzing concavity and inflection points : Analyzing functions Second derivative test : Analyzing functions Sketching curves : Analyzing functions Connecting f, f', and f'' : Analyzing functions Solving optimization problems : Analyzing functions Analyzing implicit relations : Analyzing functions Calculator-active practice : Analyzing functions.

Parametric equations, polar coordinates, and vector-valued functions. Parametric equations intro : Parametric equations, polar coordinates, and vector-valued functions Second derivatives of parametric equations : Parametric equations, polar coordinates, and vector-valued functions Vector-valued functions : Parametric equations, polar coordinates, and vector-valued functions. Planar motion : Parametric equations, polar coordinates, and vector-valued functions Polar functions : Parametric equations, polar coordinates, and vector-valued functions.

Course challenge.



0コメント

  • 1000 / 1000