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 5.2 - Differentiating Sums of Functions

which is a sum of two functions of x, Therefore, if What would be? The answer is that the derivative is the sum of the derivatives of the two functions To prove this let us return to the definition of the derivative.

We can express a small change in f, , equal to . Therefore:

and taking the limit as goes to zero gives us the instantaneous rate of change of f with respect to x.

Combing the A(x) and B(x) terms together simplifies the above expression to:

Which reduces to:

Therefore if f(x) is a sum of two functions of x, then its derivative with respect to x is the sum of the derivatives of the functions with respect to x. Thus:

Similarly if f(x) is defined in terms of a difference among some functions of x, then is the sum of the difference among the derivatives of the functions.

Next section -> Section 5.3 - Differentiating Products of Functions

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