Continuity and Differentiability — Class 12 Maths Notes

Topics covered in Continuity and Differentiability

1. 1.Introduction to Continuity and Differentiability

   Introduction to Continuity and Differentiability

   Imagine you're drawing a line without lifting your pencil off the paper. As long as the line flows smoothly without any breaks or sharp corners, we can say the drawing is continuous. In mathematics, continuity is a similar concept where we check if a function has no breaks or holes in its graph. This means you can draw the graph from start to finish without lifting your pencil.

   Now, think about driving on a road. Differentiability in math is like driving on a road where you can smoothly steer without any sudden turns. A function is differentiable if you can find a unique tangent at every point on its graph. This tangent gives you the slope of the function at that point, which tells you how steeply the function is changing.

   Why Are These Concepts Important?

   1. Real-Life Applications:

      • Continuity helps in ensuring that there are no sudden jumps in real-world measurements, which is crucial in fields like engineering and physics.
      • Differentiability allows us to calculate rates of change in various contexts, such as speed in mechanics or growth rates in biology.

   2. Career Applications:

      • Economists use these concepts to model economic growth and predict changes in markets.
      • In the tech industry, algorithms that adjust continuously based on user data rely on principles of continuity and differentiability.

   Hands-on Activity

   To grasp these concepts better, try this simple activity:

   • Take a function, say 𝑓(𝑥)=𝑥2f(x)=x2, and draw its graph on a piece of paper.
   • Check for continuity by seeing if you can draw it without lifting your pencil.
   • Then, check for differentiability by drawing tangent lines at different points on the graph.

2. 2.Continuity

   Understanding Continuity with a Simple Example

   Imagine you're drawing a line on a piece of paper without lifting your pen. As long as your pen stays on the paper and the line has no breaks or gaps, the line is continuous. Now, let's relate this to a mathematical function.

   What is Continuity?

   In mathematics, a function is said to be continuous at a point if you can draw the function's graph at that point without lifting your pen. In simpler terms, a continuous function has no holes, jumps, or breaks in its graph at that point. It's like smoothly following a path without any interruptions.

   Mathematical Definition

   A function 𝑓(𝑥)f(x) is considered continuous at a point 𝑥=𝑎x=a if the following three conditions are met:

   1. Function is Defined at 𝑎a: The value 𝑓(𝑎)f(a) must exist.
   2. Limit Exists at 𝑎a: The limit as 𝑥x approaches 𝑎a from both sides, lim⁡𝑥→𝑎𝑓(𝑥)limx→a​f(x), must exist.
   3. Limit Equals the Function Value: The limit as 𝑥x approaches 𝑎a must equal the function value at 𝑎a, i.e., lim⁡𝑥→𝑎𝑓(𝑥)=𝑓(𝑎)limx→a​f(x)=f(a).

   These three conditions ensure that there are no sudden changes or undefined points at 𝑥=𝑎x=a on the graph.

   Example: A Continuous Function

   Consider the function 𝑓(𝑥)=𝑥2f(x)=x2. This function is continuous because:

   • At any point 𝑎a, 𝑓(𝑎)=𝑎2f(a)=a2 is defined.
   • The limit as 𝑥x approaches any point 𝑎a also equals 𝑎2a2.
   • Therefore, lim⁡𝑥→𝑎𝑥2=𝑎2=𝑓(𝑎)limx→a​x2=a2=f(a), which matches the value of the function at that point.

   Real-World Example of Continuity

   Imagine filling a balloon with air. As the air goes in, the size of the balloon increases smoothly and continuously. There's no abrupt jump in the volume of the balloon as long as it's filling; it expands uniformly. This represents a continuous change, similar to how a continuous function behaves.

   Why is Continuity Important?

   Continuity in functions is crucial because it allows us to predict and understand behavior around certain points. For instance, in physics, understanding the continuous nature of motion helps in predicting the position and speed of objects at any given time.

   Real-Life Applications

   • Engineering: Ensuring bridges and roads are smoothly connected without abrupt elevations or drops.
   • Meteorology: Predicting weather changes smoothly over time, helping in more accurate forecasts.
   • Economics: Analyzing stock market trends where prices change continuously under normal conditions.

   Conclusion

   Continuity ensures smooth transitions and predictable behavior in mathematical functions, which is essential for various practical applications in science, engineering, and economics.

3. 3.Algebra of Continuous Functions

   Algebra of Continuous Functions

   When studying continuous functions, it's not just about understanding a single function's behavior but also how these functions behave when combined through various operations like addition, subtraction, multiplication, and division. This study leads us into the algebra of continuous functions.

   Definitions

   First, let's define a continuous function: A function 𝑓(𝑥)f(x) is continuous at a point 𝑥=𝑎x=a if lim⁡𝑥→𝑎𝑓(𝑥)=𝑓(𝑎)limx→a​f(x)=f(a). A function is continuous over an interval if it is continuous at every point within that interval.

   Theorems of Algebra of Continuous Functions

   1. Sum of Continuous Functions: If 𝑓(𝑥)f(x) and 𝑔(𝑥)g(x) are continuous at 𝑥=𝑎x=a, then ℎ(𝑥)=𝑓(𝑥)+𝑔(𝑥)h(x)=f(x)+g(x) is also continuous at 𝑥=𝑎x=a.

   2. Product of Continuous Functions: If 𝑓(𝑥)f(x) and 𝑔(𝑥)g(x) are continuous at 𝑥=𝑎x=a, then ℎ(𝑥)=𝑓(𝑥)⋅𝑔(𝑥)h(x)=f(x)⋅g(x) is also continuous at 𝑥=𝑎x=a.

   3. Quotient of Continuous Functions: If 𝑓(𝑥)f(x) and 𝑔(𝑥)g(x) are continuous at 𝑥=𝑎x=a and 𝑔(𝑎)≠0g(a)=0, then ℎ(𝑥)=𝑓(𝑥)𝑔(𝑥)h(x)=g(x)f(x)​ is continuous at 𝑥=𝑎x=a.

   4. Composite of Continuous Functions: If 𝑓(𝑥)f(x) is continuous at 𝑥=𝑎x=a and 𝑔(𝑥)g(x) is continuous at 𝑥=𝑓(𝑎)x=f(a), then the composition ℎ(𝑥)=𝑔(𝑓(𝑥))h(x)=g(f(x)) is continuous at 𝑥=𝑎x=a.

   Examples

   1. Sum of Functions

      • Consider 𝑓(𝑥)=𝑥2f(x)=x2 and 𝑔(𝑥)=𝑥3g(x)=x3, both of which are continuous everywhere. Their sum ℎ(𝑥)=𝑥2+𝑥3h(x)=x2+x3 is also continuous everywhere because the sum of two continuous functions is continuous.

   2. Product of Functions

      • Let 𝑓(𝑥)=sin⁡(𝑥)f(x)=sin(x) and 𝑔(𝑥)=cos⁡(𝑥)g(x)=cos(x), both continuous everywhere on the real line. The product ℎ(𝑥)=sin⁡(𝑥)⋅cos⁡(𝑥)h(x)=sin(x)⋅cos(x) is continuous everywhere.

   3. Quotient of Functions

      • If 𝑓(𝑥)=𝑥2+1f(x)=x2+1 and 𝑔(𝑥)=𝑥+1g(x)=x+1 (note 𝑔(𝑥)≠0g(x)=0 for 𝑥≠−1x=−1), the quotient ℎ(𝑥)=𝑥2+1𝑥+1h(x)=x+1x2+1​ is continuous everywhere except at 𝑥=−1x=−1.

   4. Composite of Functions

      • Consider 𝑓(𝑥)=𝑥2f(x)=x2 (continuous everywhere) and 𝑔(𝑥)=𝑥g(x)=x​ (continuous for 𝑥≥0x≥0). The composite function ℎ(𝑥)=𝑥2h(x)=x2​ simplifies to ℎ(𝑥)=∣𝑥∣h(x)=∣x∣, which is continuous everywhere.

   Applications and Importance

   Understanding the algebra of continuous functions is crucial for solving complex problems in calculus, particularly in areas involving limits and integration. These properties ensure that functions retain continuity through various operations, which is essential for accurate modeling in physics, engineering, and economics.

   This algebraic framework allows us to predict and analyze the behavior of complex systems, ensuring smooth transitions and avoiding unexpected disruptions in real-world applications.

4. 4.Differentiability

   Differentiability

   Differentiability is a concept in calculus that deals with the existence and nature of a derivative at a given point in a function. It provides a way to measure how a function changes at a particular point and is fundamental to understanding rates of change, optimization, and many other aspects of mathematical analysis.

   Definition of Differentiability

   A function 𝑓(𝑥)f(x) is said to be differentiable at a point 𝑥=𝑎x=a if the following limit exists:

   lim⁡ℎ→0𝑓(𝑎+ℎ)−𝑓(𝑎)ℎlimhf(a+h)−f(a)​

   This limit, if it exists, is defined as the derivative of 𝑓f at 𝑥=𝑎x=a, denoted 𝑓′(𝑎)f′(a). This derivative represents the slope of the tangent line to the function at that point.

   Theorem: Differentiability Implies Continuity

   Statement: If a function is differentiable at a point 𝑥=𝑎x=a, then it is also continuous at that point.

   Proof: To prove this theorem, we start by assuming that 𝑓(𝑥)f(x) is differentiable at 𝑥=𝑎x=a. By the definition of differentiability, the following limit exists and is finite:

   lim⁡ℎ→0𝑓(𝑎+ℎ)−𝑓(𝑎)ℎ=𝐿limhf(a+h)−f(a)​=L

   where 𝐿L is a real number. We need to show that 𝑓(𝑥)f(x) is continuous at 𝑥=𝑎x=a, which means:

   lim⁡𝑥→𝑎𝑓(𝑥)=𝑓(𝑎)limf(x)=f(a)

   Rewrite 𝑓(𝑥)f(x) as:

   𝑓(𝑎+ℎ)=𝑓(𝑎)+ℎ⋅𝑓(𝑎+ℎ)−𝑓(𝑎)ℎf(a+h)=f(a)+h⋅hf(a+h)−f(a)​

   Taking the limit as ℎ→0h→0, we have:

   lim⁡ℎ→0𝑓(𝑎+ℎ)=𝑓(𝑎)+lim⁡ℎ→0ℎ⋅𝑓(𝑎+ℎ)−𝑓(𝑎)ℎlimf(a+h)=f(a)+limh⋅hf(a+h)−f(a)​

   Since the limit of the derivative exists (by differentiability), and ℎh approaches zero, the product ℎ⋅𝑓(𝑎+ℎ)−𝑓(𝑎)ℎh⋅hf(a+h)−f(a)​ approaches zero. Thus:

   lim⁡ℎ→0𝑓(𝑎+ℎ)=𝑓(𝑎)limf(a+h)=f(a)

   This confirms that 𝑓(𝑥)f(x) is continuous at 𝑥=𝑎x=a.

   Examples Based on Differentiability

   1. Differentiability of 𝑓(𝑥)=𝑥2f(x)=x2 at 𝑥=0x=0

      • Function: 𝑓(𝑥)=𝑥2f(x)=x2
      • To find if it's differentiable at 𝑥=0x=0, calculate:lim⁡ℎ→0𝑓(0+ℎ)−𝑓(0)ℎ=lim⁡ℎ→0ℎ2ℎ=lim⁡ℎ→0ℎ=0limhf(0+h)−f(0)​=limhh2​=limh=0
      • Since the limit exists, 𝑓(𝑥)=𝑥2f(x)=x2 is differentiable at 𝑥=0x=0, and the derivative is 0.

   2. Differentiability of 𝑓(𝑥)=∣𝑥∣f(x)=∣x∣ at 𝑥=0x=0

      • Function: 𝑓(𝑥)=∣𝑥∣f(x)=∣x∣
      • To find if it's differentiable at 𝑥=0x=0, calculate:lim⁡ℎ→0+∣ℎ∣−∣0∣ℎ=lim⁡ℎ→0+ℎℎ=1limh∣h∣−∣0∣​=limhh​=1lim⁡ℎ→0−∣ℎ∣−∣0∣ℎ=lim⁡ℎ→0−−ℎℎ=−1limh∣h∣−∣0∣​=limh−h​=−1
      • Since the limits from the right and the left are not equal, 𝑓(𝑥)=∣𝑥∣f(x)=∣x∣ is not differentiable at 𝑥=0x=0.

   These examples illustrate how the concept of differentiability can be applied to understand the behavior of functions at specific points and how they can affect the smoothness of the function's graph. Differentiability is a key concept in calculus, used extensively in the sciences and engineering to model smooth and predictable changes.

5. 5.Derivatives of Composite Functions

   Derivatives of Composite Functions
   

   The derivative of a composite function, commonly known by the chain rule, is a fundamental concept in calculus. This rule allows us to find the derivative of a composite function efficiently.

   Definition

   Given two functions 𝑓f and 𝑔g, where 𝑦=𝑓(𝑢)y=f(u) and 𝑢=𝑔(𝑥)u=g(x), the composite function 𝑦=𝑓(𝑔(𝑥))y=f(g(x)) represents the composition of 𝑓f and 𝑔g. The chain rule helps us differentiate 𝑦=𝑓(𝑔(𝑥))y=f(g(x)) with respect to 𝑥x.

   Theorem: Chain Rule

   Statement: If 𝑓f is differentiable at 𝑢=𝑔(𝑥)u=g(x) and 𝑔g is differentiable at 𝑥x, then the composite function 𝑓(𝑔(𝑥))f(g(x)) is differentiable at 𝑥x, and its derivative is given by:

   𝑑𝑦𝑑𝑥=𝑑𝑓𝑑𝑢⋅𝑑𝑢𝑑𝑥dxdy​=dudf​⋅dxdu​

   where 𝑑𝑓𝑑𝑢dudf​ is the derivative of 𝑓f with respect to 𝑢u, and 𝑑𝑢𝑑𝑥dxdu​ is the derivative of 𝑔g with respect to 𝑥x.

   Proof: Given 𝑦=𝑓(𝑔(𝑥))y=f(g(x)),

   1. Let 𝑢=𝑔(𝑥)u=g(x). By differentiability, 𝑑𝑢=𝑔′(𝑥)𝑑𝑥du=g′(x)dx.
   2. The change in 𝑦y due to a small change 𝑑𝑢du in 𝑢u is 𝑑𝑦=𝑓′(𝑢)𝑑𝑢dy=f′(u)du.
   3. Substituting for 𝑑𝑢du, we get 𝑑𝑦=𝑓′(𝑢)𝑔′(𝑥)𝑑𝑥dy=f′(u)g′(x)dx.
   4. Hence, the derivative 𝑑𝑦𝑑𝑥=𝑓′(𝑔(𝑥))𝑔′(𝑥)dxdy​=f′(g(x))g′(x).

   Examples

   1. Example 1: Derivative of sin⁡(𝑥2)sin(x2)

      • Here, 𝑓(𝑢)=sin⁡(𝑢)f(u)=sin(u) and 𝑢=𝑔(𝑥)=𝑥2u=g(x)=x2.
      • Using the chain rule, 𝑑𝑓𝑑𝑢=cos⁡(𝑢)dudf​=cos(u) and 𝑑𝑢𝑑𝑥=2𝑥dxdu​=2x.
      • Therefore, 𝑑𝑑𝑥[sin⁡(𝑥2)]=cos⁡(𝑥2)⋅2𝑥dxd​[sin(x2)]=cos(x2)⋅2x.

   2. Example 2: Derivative of 𝑒3𝑥e3x

      • Here, 𝑓(𝑢)=𝑒𝑢f(u)=eu and 𝑢=𝑔(𝑥)=3𝑥u=g(x)=3x.
      • Using the chain rule, 𝑑𝑓𝑑𝑢=𝑒𝑢dudf​=eu and 𝑑𝑢𝑑𝑥=3dxdu​=3.
      • Therefore, 𝑑𝑑𝑥[𝑒3𝑥]=𝑒3𝑥⋅3dxd​[e3x]=e3x⋅3.

   These examples demonstrate how to apply the chain rule to compute the derivatives of composite functions. The chain rule is essential in many areas of calculus, especially when dealing with functions where one function is nested inside another. It simplifies the differentiation process and ensures accurate results in calculating rates of change in more complex situations.

6. 6.Derivatives of implicit functions

   Derivatives of Implicit Functions

   When dealing with implicit functions, the relationship between variables is expressed not as one variable explicitly in terms of another but through an equation involving both variables. Differentiating implicit functions often requires a technique known as implicit differentiation.

   Definition

   An implicit function is a function where 𝑦y is defined implicitly in terms of 𝑥x through an equation like 𝐹(𝑥,𝑦)=0F(x,y)=0. For example, the equation 𝑥2+𝑦2=1x2+y2=1 defines 𝑦y implicitly as a function of 𝑥x.

   Implicit Differentiation

   Implicit differentiation is the process used to find the derivative of 𝑦y with respect to 𝑥x when 𝑦y is defined implicitly by an equation involving 𝑥x and 𝑦y. This method relies on applying the chain rule to differentiate both sides of the equation with respect to 𝑥x.

   Steps to Differentiate an Implicit Function:

   1. Differentiate both sides of the equation with respect to 𝑥x, treating 𝑦y as a function of 𝑥x.
   2. Every time you differentiate a term involving 𝑦y, apply the chain rule: 𝑑𝑑𝑥[𝑦𝑛]=𝑛𝑦𝑛−1𝑑𝑦𝑑𝑥dxd​[yn]=nyn−1dxdy​.
   3. Solve the resulting equation for 𝑑𝑦𝑑𝑥dxdy​ to find the derivative.

   Example of Implicit Differentiation

   Example: Differentiate the circle equation 𝑥2+𝑦2=1x2+y2=1 with respect to 𝑥x.

   1. Differentiate Each Term:
      𝑑𝑑𝑥[𝑥2]=2𝑥dxd​[x2]=2x 𝑑𝑑𝑥[𝑦2]=2𝑦𝑑𝑦𝑑𝑥dxd​[y2]=2ydxdy​ (using the chain rule since 𝑦y is a function of 𝑥x)

   2. Write the Differentiated Equation:
      2𝑥+2𝑦𝑑𝑦𝑑𝑥=02x+2ydxdy​=0

   3. Solve for 𝑑𝑦𝑑𝑥dxdy​:
      2𝑦𝑑𝑦𝑑𝑥=−2𝑥2ydxdy​=−2x 𝑑𝑦𝑑𝑥=−𝑥𝑦dxdy​=−yx​

   This gives us the slope of the tangent to the circle at any point (𝑥,𝑦)(x,y) on the circle.

   Applications of Implicit Differentiation

   Implicit differentiation is used widely in calculus, particularly in:

   • Geometry: To find the slopes of tangent lines to curves that are not easily expressed as functions of 𝑥x or 𝑦y.
   • Engineering: In situations involving two or more variables where variables are related in complex ways that are not easy to separate.
   • Economics: In optimization problems where the relationship between variables is defined implicitly.

   Conclusion

   Implicit differentiation is a powerful tool in calculus that helps us handle cases where the relationship between variables isn’t straightforward. It extends our ability to apply calculus to a broader range of problems, enhancing understanding and solutions in multiple fields of study.

7. 7.Exponential and Logarithmic Functions

   Exponential and Logarithmic Functions

   Exponential and logarithmic functions are fundamental in mathematics, playing crucial roles in various applications across science, engineering, economics, and more.

   Definitions

   Exponential Functions: An exponential function is defined as 𝑓(𝑥)=𝑎𝑥f(x)=ax, where 𝑎a is a positive constant known as the base, and 𝑥x is the exponent. The function grows or decays exponentially based on the value of 𝑎a.

   Logarithmic Functions: A logarithmic function is the inverse of an exponential function. It is defined as 𝑔(𝑥)=log⁡𝑎(𝑥)g(x)=loga​(x), where 𝑎a is the base and satisfies 𝑎>0a>0 and 𝑎≠1a=1, and 𝑥x is the argument of the logarithm. The logarithmic function answers the question: "To what exponent must the base 𝑎a be raised, to produce the number 𝑥x?"

   Theorems and Properties

   Theorem 1: Exponential Growth/Decay

   • Statement: If 𝑓(𝑥)=𝑎𝑥f(x)=ax where 𝑎>1a>1, the function represents exponential growth; if 0<𝑎<10<a<1, it represents exponential decay.
   • Proof:
     • Growth: If 𝑎>1a>1, then 𝑎𝑥ax increases as 𝑥x increases because each increment in 𝑥x multiplies the function’s output by 𝑎a, which is greater than 1.
     • Decay: If 0<𝑎<10<a<1, then 𝑎𝑥ax decreases as 𝑥x increases because each increment in 𝑥x multiplies the function’s output by 𝑎a, which is less than 1.

   Theorem 2: The Inverse Nature of Logarithmic and Exponential Functions

   • Statement: The functions 𝑓(𝑥)=𝑎𝑥f(x)=ax and 𝑔(𝑥)=log⁡𝑎(𝑥)g(x)=loga​(x) are inverses of each other.
   • Proof:
     • Given 𝑦=𝑎𝑥y=ax, taking the logarithm base 𝑎a of both sides gives log⁡𝑎(𝑦)=𝑥loga​(y)=x. Hence, if 𝑦=𝑎𝑥y=ax, then 𝑥=log⁡𝑎(𝑦)x=loga​(y), and vice versa.

   Examples Based on These Concepts

   Example 1: Exponential Growth

   • Scenario: A population of bacteria doubles every hour. If you start with 1 bacterium, the population after 𝑡t hours can be modeled by 𝑃(𝑡)=2𝑡P(t)=2t.
   • Analysis: At 𝑡=0t=0, 𝑃(0)=1P(0)=1. At 𝑡=1t=1, 𝑃(1)=2P(1)=2. At 𝑡=2t=2, 𝑃(2)=4P(2)=4, and so on. This is a classic example of exponential growth.

   Example 2: Logarithmic Function Application

   • Scenario: You need to find the time 𝑡t it takes for an investment to quadruple in value if it grows continuously at an annual rate of 5%. Using the formula 𝐴=𝑃𝑒𝑟𝑡A=Pert, where 𝐴A is the amount after 𝑡t years, 𝑃P is the principal, and 𝑟r is the rate.
   • Analysis: To find the time 𝑡t when the investment is four times the original (i.e., 𝐴=4𝑃A=4P), set up the equation 4𝑃=𝑃𝑒0.05𝑡4P=Pe0.05t. Simplify to 4=𝑒0.05𝑡4=e0.05t, then take the natural logarithm of both sides: ln⁡(4)=0.05𝑡ln(4)=0.05t. Solving for 𝑡t, 𝑡=ln⁡(4)0.05t=0.05ln(4)​.

8. 8.Logarithmic Differentiation

   Logarithmic Differentiation

   Logarithmic differentiation is a technique in calculus that helps in finding derivatives of functions, particularly when the functions are products, quotients, or powers of functions whose derivatives are not straightforward to calculate using standard rules of differentiation. This method involves taking the logarithm of a function, simplifying using the properties of logarithms, and then differentiating using the chain rule.

   Why Use Logarithmic Differentiation?

   • It simplifies the differentiation of complex functions where direct application of differentiation rules would be cumbersome.
   • It is especially useful for functions involving variables raised to variable powers or products and quotients of several functions.

   Steps of Logarithmic Differentiation

   1. Take the natural logarithm (ln) of both sides of the equation 𝑦=𝑓(𝑥)y=f(x).
   2. Use logarithmic identities to simplify the expression. This might involve turning products into sums, quotients into differences, and powers into products.
   3. Differentiate both sides with respect to 𝑥x, applying the chain rule as necessary.
   4. Solve for 𝑑𝑦𝑑𝑥dxdy​ by isolating 𝑑𝑦𝑑𝑥dxdy​ on one side of the equation.

   Example 1: Differentiating 𝑦=𝑥𝑥y=xx

   1. Take the natural logarithm of both sides:ln⁡𝑦=ln⁡(𝑥𝑥)lny=ln(xx)
   2. Simplify using logarithmic properties:ln⁡𝑦=𝑥ln⁡𝑥lny=xlnx
   3. Differentiate both sides with respect to 𝑥x:1𝑦𝑑𝑦𝑑𝑥=ln⁡𝑥+𝑥⋅1𝑥=ln⁡𝑥+1y1​dxdy​=lnx+x⋅x1​=lnx+1
   4. Solve for 𝑑𝑦𝑑𝑥dxdy​:𝑑𝑦𝑑𝑥=𝑦(ln⁡𝑥+1)=𝑥𝑥(ln⁡𝑥+1)dxdy​=y(lnx+1)=xx(lnx+1)

   Example 2: Differentiating 𝑦=𝑥2sin⁡(𝑥)y=sin(x)x2​

   1. Take the natural logarithm of both sides:ln⁡𝑦=ln⁡(𝑥2sin⁡(𝑥))lny=ln(sin(x)x2​)
   2. Simplify using logarithmic properties:ln⁡𝑦=ln⁡𝑥2−ln⁡sin⁡(𝑥)=2ln⁡𝑥−ln⁡sin⁡(𝑥)lny=lnx2−lnsin(x)=2lnx−lnsin(x)
   3. Differentiate both sides with respect to 𝑥x:1𝑦𝑑𝑦𝑑𝑥=2𝑥−cos⁡(𝑥)sin⁡(𝑥)y1​dxdy​=x2​−sin(x)cos(x)​
   4. Solve for 𝑑𝑦𝑑𝑥dxdy​:𝑑𝑦𝑑𝑥=𝑦(2𝑥−cot⁡(𝑥))=𝑥2sin⁡(𝑥)(2𝑥−cot⁡(𝑥))dxdy​=y(x2​−cot(x))=sin(x)x2​(x2​−cot(x))

   These examples illustrate how logarithmic differentiation can manage complex differentiation tasks by breaking them down into simpler steps. This method leverages the properties of logarithms to simplify the product, quotient, and power rules into more manageable linear operations, making differentiation more straightforward.