Understanding Calculus

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  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 5.4 - Differentiating Functions of any power N

We have proven that if then for n equal to a positive integer, i.e. 1, 2, 3, 4 etc. What if n were a fraction such that . As we shall prove, the derivative of any function of x of the form is always where n is any real number, positive, negative, or fraction. This is no coincidence but is because of the way exponents are defined as a continuous operation for any n. Let us consider the first case of , where n is any positive integer and 1/n is a fraction.

Raising both sides to the níth power:

Differentiating x, with respect to f yields:

Taking the reciprocal of the function:

From the definition of the function, we know that . Making this substitution:

This concludes the proof that if then , for any positive n, integer or fraction. We will prove it also holds true for n as a negative number.

We can use the product rule to prove that the derivative of for all n, negative real numbers also. Consider the function:

Multiplying both sides by yields:

The left side of the equation represents a product of two functions, f and , and the right side is a constant function, . Since both functions are equal to each then their derivatives must be equal. Using the product rule to differentiate the left side with respect to x results in:

Similarly, differentiating the right side with respect to x yields:

Setting the derivatives of the left and right side equal to each other:

Remember that:

Making the required substitutions:

We can now solve for

Therefore if , where -n is a negative integer. We have proven that if for positive real numbers and negative integers. What remains is to prove it true for negative fractional powers as well. To do this let,

. Differentiating both sides with respect to x yields:

Substituting known values for f(x) and solving for gives us:

Thus if for any n, positive or negative real number. This is true because of the way exponents are defined as a uniform operation for any n. The purpose of going through all the proofs for the different cases of n was to give you a better understanding of how to differentiate functions of x with respect to x.

Next section -> Section 5.5 - Differentiating Functions of Functions

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