## 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 5.2 - 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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