derivative, in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations.
What are derivatives with example?
A derivative is an instrument whose value is derived from the value of one or more underlying, which can be commodities, precious metals, currency, bonds, stocks, stocks indices, etc. Four most common examples of derivative instruments are Forwards, Futures, Options and Swaps. 2.
In mathematics, derivative is defined as the method that shows the simultaneous rate of change. That means it is used to represent the amount by which the given function is changing at a certain point.
In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation.[1] The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
4. Find the derivative of a function at a given point
Learning Objectives
Sketch the graph of a function using the graph of its derivative
Understand the relationship between continuity and differentiability
Determine the differentiability of a function at a given point
Estimate the derivative at a given point using tables
5.
6.
7. f ‘(x) = 9x2
+ 2.
f ‘(1) = 9 + 2 = 11
f ‘(2) = 9(4) + 2 = 38
and
f ‘(3) = 9(9) + 2 = 83.
8.
9.
10. Note:
Keep in mind that,
the value of the derivative function at a point is the slope of the tangent line at that
point.
11. Rather than worrying about exact values of f (x),
we only wish to find the general shape of its
graph.
12.
13.
14.
15. The graph indicates a sharp corner at x = 2, so you might expect
that the derivative does not exist.
To verify this, we investigate the derivative by evaluating one-sided
limits.
For h > 0, note that (2 + h) > 2 and so, f (2 + h) = 2(2 + h).
Likewise, if h < 0, (2 + h) < 2 and so, f (2 + h) =
4.
16. Figures 2.19a–2.19d show a variety of functions for which f (a) does
not exist.
In each case, convince yourself that the derivative does not exist.
17. There are many times in applications when it is not possible or practical to compute derivatives symbolically.
This is frequently the case where we have only some data (i.e., a table of values) representing an otherwise
unknown function.
18. Solution The instantaneous velocity is the limit of the average velocity as the time interval shrinks.
We first compute the average velocities over the shortest intervals given, from 5.9 to 6.0 and from
6.0 to 6.1.
Since these are the best individual estimates available from the data, we could just split the
difference and estimate a velocity of 35.1 ft/s.
However, there is useful information in the rest of the data. Based on the accompanying table, we
can conjecture that the sprinter was reaching a peak speed at about the 6-second mark.
Thus, we might accept the higher estimate of 35.2 ft/s.
We should emphasize that there is not a single correct answer to this question, since the data are
incomplete (i.e., we know the distance only at fixed times, rather than over a continuum of times).
