## 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 5.1 - 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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