## Understanding Calculus

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 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

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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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