3.4 The Quotient 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

What is the derivative of $\ds (x^2+1)/(x^3-3x)$? More generally, we'd like to have a formula to compute the derivative of $f(x)/g(x)$ if we already know $f'(x)$ and $g'(x)$. Instead of attacking this problem head-on, let's notice that we've already done part of the problem: $f(x)/g(x)= f(x)\cdot(1/g(x))$, that is, this is "really'' a product, and we can compute the derivative if we know $f'(x)$ and $(1/g(x))'$. So really the only new bit of information we need is $(1/g(x))'$ in terms of $g'(x)$. As with the product rule, let's set this up and see how far we can get: $$ \eqalign{ {d\over dx}{1\over g(x)}&=\lim_{\Delta x\to0} {{1\over g(x+\Delta x)}-{1\over g(x)}\over\Delta x}\cr &=\lim_{\Delta x\to0} {{g(x)-g(x+\Delta x)\over g(x+\Delta x)g(x)}\over\Delta x}\cr &=\lim_{\Delta x\to0} {g(x)-g(x+\Delta x)\over g(x+\Delta x)g(x)\Delta x}\cr &=\lim_{\Delta x\to0} -{g(x+\Delta x)-g(x)\over \Delta x} {1\over g(x+\Delta x)g(x)}\cr &=-{g'(x)\over g(x)^2}\cr }$$ Now we can put this together with the product rule: $${d\over dx}{f(x)\over g(x)}=f(x){-g'(x)\over g(x)^2}+f'(x){1\over g(x)}={-f(x)g'(x)+f'(x)g(x)\over g(x)^2}= {f'(x)g(x)-f(x)g'(x)\over g(x)^2}. $$

Example 3.4.1 Compute the derivative of $\ds (x^2+1)/(x^3-3x)$. $${d\over dx}{x^2+1\over x^3-3x}={2x(x^3-3x)-(x^2+1)(3x^2-3)\over(x^3-3x)^2}= {-x^4-6x^2+3\over (x^3-3x)^2}. $$

$\square$

It is often possible to calculate derivatives in more than one way, as we have already seen. Since every quotient can be written as a product, it is always possible to use the product rule to compute the derivative, though it is not always simpler.

Example 3.4.2 Find the derivative of $\ds \sqrt{625-x^2}/\sqrt{x}$ in two ways: using the quotient rule, and using the product rule.

Quotient rule: $${d\over dx}{\sqrt{625-x^2}\over\sqrt{x}} = {\sqrt{x}(-x/\sqrt{625-x^2})-\sqrt{625-x^2}\cdot 1/(2\sqrt{x})\over x}.$$ Note that we have used $\ds \sqrt{x}=x^{1/2}$ to compute the derivative of $\ds \sqrt{x}$ by the power rule.

Product rule: $${d\over dx}\sqrt{625-x^2} x^{-1/2} = \sqrt{625-x^2} {-1\over 2}x^{-3/2}+{-x\over \sqrt{625-x^2}}x^{-1/2}. $$

With a bit of algebra, both of these simplify to $$-{x^2+625\over 2\sqrt{625-x^2}x^{3/2}}.$$

$\square$

Occasionally you will need to compute the derivative of a quotient with a constant numerator, like $\ds 10/x^2$. Of course you can use the quotient rule, but it is usually not the easiest method. If we do use it here, we get $${d\over dx}{10\over x^2}={x^2\cdot 0-10\cdot 2x\over x^4}= {-20\over x^3},$$ since the derivative of 10 is 0. But it is simpler to do this: $${d\over dx}{10\over x^2}={d\over dx}10x^{-2}=-20x^{-3}.$$ Admittedly, $\ds x^2$ is a particularly simple denominator, but we will see that a similar calculation is usually possible. Another approach is to remember that $${d\over dx}{1\over g(x)}={-g'(x)\over g(x)^2},$$ but this requires extra memorization. Using this formula, $${d\over dx}{10\over x^2}=10{-2x\over x^4}.$$ Note that we first use linearity of the derivative to pull the 10 out in front.

Exercises 3.4

Find the derivatives of the functions in 1–4 using the quotient rule.

Ex 3.4.1 $\ds {x^3\over x^3-5x+10}$ (answer)

Ex 3.4.2 $\ds {x^2+5x-3\over x^5-6x^3+3x^2-7x+1}$ (answer)

Ex 3.4.3 $\ds {\sqrt{x}\over\sqrt{625-x^2}}$ (answer)

Ex 3.4.4 $\ds {\sqrt{625-x^2}\over x^{20}}$ (answer)

Ex 3.4.5 Find an equation for the tangent line to $\ds f(x) = (x^2 - 4)/(5-x)$ at $x= 3$. (answer)

Ex 3.4.6 Find an equation for the tangent line to $\ds f(x) = (x-2)/(x^3 + 4x - 1)$ at $x=1$. (answer)

Ex 3.4.7 Let $P$ be a polynomial of degree $n$ and let $Q$ be a polynomial of degree $m$ (with $Q$ not the zero polynomial). Using sigma notation we can write $$P=\sum _{k=0 } ^n a_k x^k,\qquad Q=\sum_{k=0}^m b_k x^k. $$ Use sigma notation to write the derivative of the rational function $P/Q$.

Ex 3.4.8 The curve $\ds y=1/(1+x^2)$ is an example of a class of curves each of which is called a witch of Agnesi. Sketch the curve and find the tangent line to the curve at $x= 5$. (The word witch here is a mistranslation of the original Italian, as described at http://mathworld.wolfram.com/WitchofAgnesi.html and http://witchofagnesi.org/. (answer)

Ex 3.4.9 If $f'(4) = 5$, $g'(4) = 12$, $(fg)(4)= f(4)g(4)=2$, and $g(4) = 6$, compute $f(4)$ and $\ds{d\over dx}{f\over g}$ at 4. (answer)