Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Calculus → Differentiation ↓


Quotient Rule


The quotient rule is a method used in calculus to find the derivative of a function that is the quotient (or division) of two other functions. This rule is necessary when dealing with functions of the form f(x) = g(x) / h(x), where g(x) and h(x) are both functions. The challenge is that these functions can behave quite differently on their own, making calculating their derivatives a little more complicated.

What is the quotient rule?

Mathematically, the quotient rule states that if you have a function given by f(x) = g(x) / h(x), then the derivative f'(x) is given by:

(f(x))' = (g'(x) * h(x) - g(x) * h'(x)) / (h(x)^2)

In simple terms, to find the derivative of a quotient:

  • Differentiate the numerator to get g'(x).
  • Multiply this derivative by the denominator h(x).
  • Differentiate the denominator to get h'(x).
  • Multiply this derivative by the numerator g(x).
  • Subtract the second product from the first product.
  • Finally, divide it by the square of the denominator h(x)^2.

Breaking the quotient rule

Let us analyse the components of the Quotient Rule:

  • g'(x) * h(x): This is the derivative of the numerator and the original denominator.
  • g(x) * h'(x): This is the value of the original numerator multiplied by the derivative of the denominator.
  • h(x)^2: This is the square of the original denominator.

In short, the rule is making sure that we take into account how both functions change, which affects the overall rate of change of the quotient.

Visual example 1: Applying the quotient rule

f(x) = x^2 / (3x + 2) (f(x))' = ((2x) * (3x + 2) - (x^2) * 3) / (3x + 2)^2

Consider a function f(x) = x^2 / (3x + 2) To find the derivative f'(x):

  1. The derivative of the fraction x^2 is 2x.
  2. The derivative of the denominator 3x + 2 is 3.
  3. Apply the quotient rule:
    4. (f(x))' = ((2x) * (3x + 2) - (x^2) * 3) / (3x + 2)^2

By simplifying this expression, we get a new function that shows how the original quotient changes.

Lesson example 2: Applying the quotient rule

Let's look at a text example including steps:

Function: f(x) = (5x - 4) / (x + 1)

We need to find the derivative of f'(x):

  1. Identify g(x) = 5x - 4 and h(x) = x + 1.
  2. Differentiate g(x) to get g'(x) = 5.
  3. Differentiate h(x) to get h'(x) = 1.
  4. Apply the quotient rule:
    5. (f(x))' = ((5) * (x + 1) - (5x - 4) * 1) / (x + 1)^2
  5. Simplify the expression to find f'(x):
    6. (f(x))' = (5x + 5 - 5x + 4) / (x + 1)^2
(f(x))' = (9) / (x + 1)^2

Visual example 2: Applying the quotient rule

f(x) = sin(x) / x (f(x))' = ((cos(x) * x) - (sin(x) * 1)) / x^2

Consider a function f(x) = sin(x) / x To find the derivative f'(x):

  1. The derivative of the fraction sin(x) is cos(x).
  2. The derivative of every x is 1.
  3. Applying the quotient rule:
    4. (f(x))' = ((cos(x) * x) - (sin(x) * 1)) / x^2

The resulting expression represents the rate of change of the original function.

Practice problems

Practice is key to mastering the quotient rule. Try these problems yourself:

  1. Differentiate: f(x) = (3x^3 - x) / (2x^2 + 1)
  2. Find f'(x): f(x) = (7x + 5) / (4x^3 - x)
  3. Calculate the derivative: f(x) = e^x / ln(x)

Conclusion

The quotient rule is a powerful tool in differentiation, enabling you to handle complex functions created by division. Understanding and applying this rule requires careful attention to the order of operations and the relationship between the numerator and denominator. With practice, it becomes an automatic part of solving calculus problems involving quotients.


Grade 11 → 4.2.6


U
username
0%
completed in Grade 11

← Prev (4.2.5)
Product Rule
Next (4.3) →
Applications of Differentiation

Comments 



Quotient Rule

https://www.buddymath.com/g11/4.2.6?bmal=en