3.1 The Power Rule

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Introduction

1 Analytic Geometry

• 1. Lines
• 2. Distance Between Two Points; Circles
• 3. Functions
• 4. Shifts and Dilations

2 Instantaneous Rate of Change: The Derivative

• 1. The slope of a function
• 2. An example
• 3. Limits
• 4. The Derivative Function
• 5. Properties of Functions

3 Rules for Finding Derivatives

• 1. The Power Rule
• 2. Linearity of the Derivative
• 3. The Product Rule
• 4. The Quotient Rule
• 5. The Chain Rule

4 Transcendental Functions

• 1. Trigonometric Functions
• 2. The Derivative of $\sin x$
• 3. A hard limit
• 4. The Derivative of $\sin x$, continued
• 5. Derivatives of the Trigonometric Functions
• 6. Exponential and Logarithmic functions
• 7. Derivatives of the exponential and logarithmic functions
• 8. Implicit Differentiation
• 9. Inverse Trigonometric Functions
• 10. Limits revisited
• 11. Hyperbolic Functions

5 Curve Sketching

• 1. Maxima and Minima
• 2. The first derivative test
• 3. The second derivative test
• 4. Concavity and inflection points
• 5. Asymptotes and Other Things to Look For

6 Applications of the Derivative

• 1. Optimization
• 2. Related Rates
• 3. Newton's Method
• 4. Linear Approximations
• 5. The Mean Value Theorem

7 Integration

• 1. Two examples
• 2. The Fundamental Theorem of Calculus
• 3. Some Properties of Integrals

8 Techniques of Integration

• 1. Substitution
• 2. Powers of sine and cosine
• 3. Trigonometric Substitutions
• 4. Integration by Parts
• 5. Rational Functions
• 6. Numerical Integration
• 7. Additional exercises

9 Applications of Integration

• 1. Area between curves
• 2. Distance, Velocity, Acceleration
• 3. Volume
• 4. Average value of a function
• 5. Work
• 6. Center of Mass
• 7. Kinetic energy; improper integrals
• 8. Probability
• 9. Arc Length
• 10. Surface Area

10 Polar Coordinates, Parametric Equations

• 1. Polar Coordinates
• 2. Slopes in polar coordinates
• 3. Areas in polar coordinates
• 4. Parametric Equations
• 5. Calculus with Parametric Equations

11 Sequences and Series

• 1. Sequences
• 2. Series
• 3. The Integral Test
• 4. Alternating Series
• 5. Comparison Tests
• 6. Absolute Convergence
• 7. The Ratio and Root Tests
• 8. Power Series
• 9. Calculus with Power Series
• 10. Taylor Series
• 11. Taylor's Theorem
• 12. Additional exercises

12 Three Dimensions

• 1. The Coordinate System
• 2. Vectors
• 3. The Dot Product
• 4. The Cross Product
• 5. Lines and Planes
• 6. Other Coordinate Systems

13 Vector Functions

• 1. Space Curves
• 2. Calculus with vector functions
• 3. Arc length and curvature
• 4. Motion along a curve

14 Partial Differentiation

• 1. Functions of Several Variables
• 2. Limits and Continuity
• 3. Partial Differentiation
• 4. The Chain Rule
• 5. Directional Derivatives
• 6. Higher order derivatives
• 7. Maxima and minima
• 8. Lagrange Multipliers

15 Multiple Integration

• 1. Volume and Average Height
• 2. Double Integrals in Cylindrical Coordinates
• 3. Moment and Center of Mass
• 4. Surface Area
• 5. Triple Integrals
• 6. Cylindrical and Spherical Coordinates
• 7. Change of Variables

16 Vector Calculus

• 1. Vector Fields
• 2. Line Integrals
• 3. The Fundamental Theorem of Line Integrals
• 4. Green's Theorem
• 5. Divergence and Curl
• 6. Vector Functions for Surfaces
• 7. Surface Integrals
• 8. Stokes's Theorem
• 9. The Divergence Theorem

17 Differential Equations

• 1. First Order Differential Equations
• 2. First Order Homogeneous Linear Equations
• 3. First Order Linear Equations
• 4. Approximation
• 5. Second Order Homogeneous Equations
• 6. Second Order Linear Equations
• 7. Second Order Linear Equations, take two

18 Useful formulas

19 Introduction to Sage

• 1. Basics
• 2. Differentiation
• 3. Integration

We start with the derivative of a power function, $\ds f(x)=x^n$. Here $n$ is a number of any kind: integer, rational, positive, negative, even irrational, as in $\ds x^\pi$. We have already computed some simple examples, so the formula should not be a complete surprise: $${d\over dx}x^n = nx^{n-1}.$$ It is not easy to show this is true for any $n$. We will do some of the easier cases now, and discuss the rest later.

The easiest, and most common, is the case that $n$ is a positive integer. To compute the derivative we need to compute the following limit: $${d\over dx}x^n = \lim_{\Delta x\to0} {(x+\Delta x)^n-x^n\over \Delta x}. $$ For a specific, fairly small value of $n$, we could do this by straightforward algebra.

Example 3.1.1 Find the derivative of $\ds f(x)=x^3$. $$\eqalign{ {d\over dx}x^3 &= \lim_{\Delta x\to0} {(x+\Delta x)^3-x^3\over \Delta x}.\cr &=\lim_{\Delta x\to0} {x^3+3x^2\Delta x+3x\Delta x^2 + \Delta x^3 -x^3\over \Delta x}.\cr &=\lim_{\Delta x\to0}{3x^2\Delta x+3x\Delta x^2 + \Delta x^3\over \Delta x}.\cr &=\lim_{\Delta x\to0}3x^2+3x\Delta x + \Delta x^2 = 3x^2.\cr }$$

$\square$

The general case is really not much harder as long as we don't try to do too much. The key is understanding what happens when $\ds (x+\Delta x)^n$ is multiplied out: $$(x+\Delta x)^n=x^n + nx^{n-1}\Delta x + a_2x^{n-2}\Delta x^2+\cdots+ +a_{n-1}x\Delta x^{n-1} + \Delta x^n.$$ We know that multiplying out will give a large number of terms all of the form $\ds x^i\Delta x^j$, and in fact that $i+j=n$ in every term. One way to see this is to understand that one method for multiplying out $\ds (x+\Delta x)^n$ is the following: In every $(x+\Delta x)$ factor, pick either the $x$ or the $\Delta x$, then multiply the $n$ choices together; do this in all possible ways. For example, for $\ds (x+\Delta x)^3$, there are eight possible ways to do this: $$\eqalign{ (x+\Delta x)(x+\Delta x)(x+\Delta x)&=xxx + xx\Delta x + x\Delta x x + x\Delta x \Delta x\cr &\qquad+ \Delta x xx + \Delta xx\Delta x + \Delta x\Delta x x + \Delta x\Delta x \Delta x\cr &= x^3 + x^2\Delta x +x^2\Delta x +x\Delta x^2\cr &\quad+x^2\Delta x +x\Delta x^2 +x\Delta x^2 +\Delta x^3\cr &=x^3 + 3x^2\Delta x + 3x\Delta x^2+\Delta x^3\cr }$$ No matter what $n$ is, there are $n$ ways to pick $\Delta x$ in one factor and $x$ in the remaining $n-1$ factors; this means one term is $\ds nx^{n-1}\Delta x$. The other coefficients are somewhat harder to understand, but we don't really need them, so in the formula above they have simply been called $\ds a_2$, $\ds a_3$, and so on. We know that every one of these terms contains $\Delta x$ to at least the power 2. Now let's look at the limit: $$\eqalign{ {d\over dx}x^n &= \lim_{\Delta x\to0} {(x+\Delta x)^n-x^n\over \Delta x}\cr &=\lim_{\Delta x\to0} {x^n + nx^{n-1}\Delta x + a_2x^{n-2}\Delta x^2+\cdots+ a_{n-1}x\Delta x^{n-1} + \Delta x^n-x^n\over \Delta x}\cr &=\lim_{\Delta x\to0} {nx^{n-1}\Delta x + a_2x^{n-2}\Delta x^2+\cdots+ a_{n-1}x\Delta x^{n-1} + \Delta x^n\over \Delta x}\cr &=\lim_{\Delta x\to0} nx^{n-1} + a_2x^{n-2}\Delta x+\cdots+ a_{n-1}x\Delta x^{n-2} + \Delta x^{n-1} = nx^{n-1}.\cr }$$

Now without much trouble we can verify the formula for negative integers. First let's look at an example:

Example 3.1.2 Find the derivative of $\ds y=x^{-3}$. Using the formula, $\ds y'=-3x^{-3-1}=-3x^{-4}$. $\square$

Here is the general computation. Suppose $n$ is a negative integer; the algebra is easier to follow if we use $n=-m$ in the computation, where $m$ is a positive integer. $$\eqalign{ {d\over dx}x^n &= {d\over dx}x^{-m} = \lim_{\Delta x\to0} {(x+\Delta x)^{-m}-x^{-m}\over \Delta x}\cr &=\lim_{\Delta x\to0} { {1\over (x+\Delta x)^m} - {1\over x^m} \over \Delta x} \cr &=\lim_{\Delta x\to0} { x^m - (x+\Delta x)^m \over (x+\Delta x)^m x^m \Delta x} \cr &=\lim_{\Delta x\to0} { x^m - (x^m + mx^{m-1}\Delta x + a_2x^{m-2}\Delta x^2+\cdots+ a_{m-1}x\Delta x^{m-1} + \Delta x^m)\over (x+\Delta x)^m x^m \Delta x} \cr &=\lim_{\Delta x\to0} { -mx^{m-1} - a_2x^{m-2}\Delta x-\cdots- a_{m-1}x\Delta x^{m-2} - \Delta x^{m-1})\over (x+\Delta x)^m x^m} \cr &={ -mx^{m-1} \over x^mx^m}= { -mx^{m-1} \over x^{2m}}= -mx^{m-1-2m}= nx^{-m-1} = nx^{n-1}.\cr }$$

We will later see why the other cases of the power rule work, but from now on we will use the power rule whenever $n$ is any real number. Let's note here a simple case in which the power rule applies, or almost applies, but is not really needed. Suppose that $f(x)=1$; remember that this "1'' is a function, not "merely'' a number, and that $f(x)=1$ has a graph that is a horizontal line, with slope zero everywhere. So we know that $f'(x)=0$. We might also write $\ds f(x)=x^0$, though there is some question about just what this means at $x=0$. If we apply the power rule, we get $\ds f'(x)=0x^{-1}=0/x=0$, again noting that there is a problem at $x=0$. So the power rule "works'' in this case, but it's really best to just remember that the derivative of any constant function is zero.

Exercises 3.1

Find the derivatives of the given functions.

Ex 3.1.1 $\ds x^{100}$ (answer)

Ex 3.1.2 $\ds x^{-100}$ (answer)

Ex 3.1.3 $\displaystyle {1\over x^5}$ (answer)

Ex 3.1.4 $\ds x^\pi$ (answer)

Ex 3.1.5 $\ds x^{3/4}$ (answer)

Ex 3.1.6 $\ds x^{-9/7}$ (answer)