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 5.3 - Differentiating Products of Functions

Consider the following function:

If we let , then f(x) can be expressed as the product of the two function A(x) and B(x) such that:

We can differentiate products of functions by using the definition of the derivative. A small change in f can be written as:

Next, divide by to calculate the rate of change of f with respect to x:

Taking the limit as goes to zero gives us the instantaneous rate of change of f with respect to x, or the derivative of f(x).

From the definition of the derivative we know that:

Multiplying both sides by this infinitely small

Since both A(x) and B(x) are functions of x, then can be substituted with respectively. Note that this substitution only holds true for going to zero. We now have:

Expanding the numerator:

Canceling terms and dividing through by reduces it to:

Thus the derivative of a function f(x) that is a product of two functions of x, is simply the product of the first function and the derivative of the second function plus the product of the second function and the derivative of the first function.

Next section -> Section 5.4 - Differentiating Functions of any Power of N

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