Understanding Calculus

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  Home
  Testimonials
  Table of Contents

  Preface
  1. Why Study
  Calculus
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  Motion
  8. Understanding
  Derivative
  9. Derivative
  Approximations
  10. Integration
  Theory
  11. Understanding
  Integration
  12. Differentials

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

  Links
  Contact

 
 
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CHAPTER 5

Chapter 5 - Differentiating Functions

Section 5.1 - Differentiating Functions

Differentiation is the process of finding the rate of change of a function. We have proven that if f is a variable dependent on an independent variable x, such that then where n is a positive integer. The derivative reflects the instantaneous rate of change of the function at any value x. The derivative is also a function of x whose value is dependent on x.

Take a look at the left side of the function, By definition the derivative of a dependent variable, f, is , which is the instantaneous rate of change of f with respect to x at any condition x. The right side of the function, , represents the independent variable whose derivative is

When differentiating a function of the form , the derivative of the dependent variable is, , and the derivative of the independent variable is . Thus differentiating a function results in a new function of x, where . The derivative is called , read f prime of x, and it represents the derivative of a function of x with respect to the independent variable, x.. If , then:

gives the instantaneous rate of change of f(x) as a function of any value, x. Remember that the rate of change of a function other than a line is not constant. Its value changes as x changes.

If f(x) were equal to a constant multiplied by a function of x such as:

The derivative of f(x) would be:

Thus the derivative of f(x) with respect to x, is the constant multiplied by the derivative of the function of x, A(x).

Next section -> Section 5.2 - Differentiating Sums of Functions

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