Derivative of Inverse Function

5 Key excerpts on "Derivative of Inverse Function"

• 

  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 9 The Idea of a Derivative of a Function 9.1 Introduction

  There are certain problems in mathematics, mechanics, physics, and many other branches of science, which cannot be solved by ordinary methods of geometry or algebra alone . To solve these problems, we have to use a new branch of mathematics known as calculus. It uses not only the ideas and methods from arithmetic, geometry, algebra, coordinate geometry, trigonometry, and so on, but also the notion of limit , which is a new idea that lies at the foundation of calculus . Using this notion as a tool, the derivative of a function is defined as the limit of a particular kind.

  The idea of derivative of a function is among the most important and powerful concepts in mathematics. This concept distinguishes calculus from other branches of mathematics. It will be found that the derivative of a function is generally a new function (derived from the original function). We call it the rate function or the derivative function.

  Calculus is the mathematics of change. The immense practical power of calculus is due to its ability to describe and predict the behavior of changing quantities. We cannot even begin to answer any question related to change unless we know what changes and how it changes? Let us discuss.

  We know that

  • the area of a circle, A (r ) = πr 2 , changes with (respect to) its radius “r ”.
  • the volume of a sphere, ( ), changes with (respect to) its radius “r ”.
  • the surface area of a cube, S(l ) = 6l 2 , changes with (respect to) the length “l ” of its side.

  Consider a function y = h (x ) whose graph is a smooth curve (not a straight line). Then, the inclination “θ” of the tangent line (drawn at any point of the curve) changes from point to point on the curve. (Later on, this observation will be used to define a (new) concept, namely, “the slope of a curve

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Differential Equations and Calculus

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 12 Derivative The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity. Conversely, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends. The derivative of a function at a chosen input value describes the best linear approx-imation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function. ________________________ WORLD TECHNOLOGIES ________________________ The process of finding a derivative is called differentiation . The reverse process is called antidifferentiation . The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation and the derivative At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  The function f is constant on [0 , 1], which is consistent with the fact that f 0 ( x ) = 0 for these x . Here’s another potential problem. The four conditions on the previous page all require that the domain be an interval like ( a, b ). What if the domain isn’t in one piece? Unfortunately, then the conclusion can totally fail to hold. For example, if f ( x ) = tan( x ), then f 0 ( x ) = sec 2 ( x ), which can’t be negative; however, you can see from the graph that y = tan( x ) fails the horizontal line test pretty miserably. (See Section 10.2.3 below to remind yourself about the graph of y = tan( x ).) So the methods of the previous section won’t work, in general, when your function has discontinuities or vertical asymptotes. 204 • Inverse Functions and Inverse Trig Functions 10.1.3 Finding the derivative of an inverse function If you know that a function f has an inverse, which we’ll call f -1 as usual, then what’s the derivative of that inverse? Here’s how you find it. Start off with the equation y = f -1 ( x ). You can rewrite this as f ( y ) = x . Now differentiate implicitly with respect to x to get d dx ( f ( y )) = d dx ( x ) . The right-hand side is easy: it’s just 1. To find the left-hand side, we use implicit differentiation (see Chapter 8). If we set u = f ( y ), then by the chain rule (noting that du/dy = f 0 ( y )), we have d dx ( f ( y )) = d dx ( u ) = du dy dy dx = f 0 ( y ) dy dx . Now divide both sides by f 0 ( y ) to get the following principle: if y = f -1 ( x ) , then dy dx = 1 f 0 ( y ) . If you want to express everything in terms of x , then you have to replace y by f -1 ( x ) to get d dx ( f -1 ( x )) = 1 f 0 ( f -1 ( x )) . In words, this means that the derivative of the inverse is basically the recipro-cal of the derivative of the original function, except that you have to evaluate this latter derivative at f -1 ( x ) instead of x .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  implicitly defined functions • Implicit differentiation technique • Parametric curves defined implicitly: Related rates 3.9 Inverse Functions and Their Derivatives • Definition of an inverse function • One-to-One property of invertible functions • Horizontal line test • Computing inverse functions • Ranges of inverse trigonometric functions: y = sin -1 x: - π 2 ≤ y ≤ π 2 y = cos -1 x: 0 ≤ y ≤ π y = tan -1 x: - π 2 < y < π 2 • Derivatives of inverse trigonometric functions: d dx [sin -1 x] = 1 √ 1-x 2 d dx [cos -1 x] = - 1 √ 1-x 2 d dx [tan -1 x] = 1 1+x 2 3.10 Logarithmic Functions and Their Deriva- tives • Logarithmic functions as inverses of exponen- tial functions • Domain and range of logarithmic functions • Graphs and graphical properties of logarith- mic functions • Inverse (cancellation) properties of logarithms: a log a x = x, log a a x = x • Laws of logarithms • Derivative of the natural logarithm function: d dx [ln x] = 1 x • Logarithmic differentiation • Derivatives of general exponential and loga- rithmic functions: d dx [a x ] = a x ln a d dx [log a x] = 1 x ln a 208 CHAPTER 3. THE DERIVATIVE Chapter 3 Review Exercises Exercises 1–16 Answer True or False. 1. If the average rate of change for f on the interval [0, h] is h + 2 and f is differentiable at 0, then f 0 (0) = 2. 2. If a function is linear, then its instantaneous rate of change is constant. 3. If f 0 (2) exists, then lim x→2 - f (x) = f (2). 4. If f is continuous at a, then f has a tangent line at (a, f (a)). 5. An equation of the tangent line to the graph of a dif- ferentiable function f at a point (a, f (a)) is given by y - f (a) = f 0 (x)(x - a). 6. The derivative of a sum of two differentiable functions is equal to the sum of their derivatives. 7. The derivative of the product of two differentiable func- tions is equal to the product of their derivatives.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  A Passage to Modern Analysis

  • William J. Terrell(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 5 The Derivative This chapter presents the basic properties of the derivative of a real valued function of a real variable. We assume the reader has experience from introductory calculus courses with the geometric idea of the derivative at an interior point of the domain as the slope of the tangent line to the graph of a function. We also assume familiar-ity with the elementary functions (polynomials, rational functions, trigonometric functions, exponential functions, logarithmic functions, and their inverses) and we shall use facts about derivatives of elementary functions in examples. For reference, we note that the natural logarithm function, denoted log( x ) in this book, is defined in Theorem 6.7.9. The exponential function exp( x ) = e x (the inverse of the natural logarithm function), as well as exponential and logarithm functions for other bases b > 0, and the sine and cosine functions, are defined and discussed in detail in Section 7.5. 5.1. The Derivative: Definition and Properties Geometrically, the slope of a function graph at a point ( a, f ( a )) on the graph indicates the rate of change of the function with respect to the independent variable as that variable approaches the point a . This rate of change is the limiting value of the slopes f ( x ) − f ( a ) x − a of chords joining the points ( x, f ( x )) and ( a, f ( a )), as x approaches a , when this limiting value exists. This limit process makes sense whenever the point a is an interior point of the domain. Definition 5.1.1. Let D be an interval of real numbers, let f : D → R , and suppose a ∈ D is an interior point. If the limit lim x → a f ( x ) − f ( a ) x − a 121 122 5. The Derivative exists, then f is said to be differentiable at a , and the limiting value is denoted by f ( a ) and called the derivative of f at a . If D is an open interval, and if f is differentiable at every a ∈ D , then we say f is differentiable on D .

  Sign up to read

  Learn more about book