Derivatives - Derivative of ln(x)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
• Let’s understand this concept with some examples.

Example 1:

Find the derivative of ln(x).

• Solution:
  • Using the formula, we find
  • Therefore, the derivative of ln(x) is .

Example 2:

Find the derivative of ln(4x).

• Solution:
  • Using the chain rule, we can differentiate ln(4x).
  • Applying the chain rule, we get
  • Simplifying further, we find .

Example 3:

Find the derivative of ln(3x^2).

• Solution:
  • Using the chain rule, we can differentiate ln(3x^2).
  • Applying the chain rule and power rule, we get
  • Simplifying further, we find .

11. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 4:

Find the derivative of ln(5x^3).

• Solution:
  • Using the chain rule and power rule, we can differentiate ln(5x^3).
  • Applying the chain rule and power rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 5:

Find the derivative of ln(2x + 1).

• Solution:
  • Using the chain rule, we can differentiate ln(2x + 1).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

12. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 6:

Find the derivative of ln(1/x).

• Solution:
  • Using the chain rule, we can differentiate ln(1/x).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 7:

Find the derivative of ln(√x).

• Solution:
  • Using the chain rule, we can differentiate ln(√x).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

21. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 8:

Find the derivative of ln(e^x).

• Solution:
  • Using the chain rule, we can differentiate ln(e^x).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 9:

Find the derivative of ln(x^2 + 1).

• Solution:
  • Using the chain rule, we can differentiate ln(x^2 + 1).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

22. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 10:

Find the derivative of ln(e^(2x)).

• Solution:
  • Using the chain rule, we can differentiate ln(e^(2x)).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 11:

Find the derivative of ln(sin(x)).

• Solution:
  • Using the chain rule, we can differentiate ln(sin(x)).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

23. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 12:

Find the derivative of ln(cos(x)).

• Solution:
  • Using the chain rule, we can differentiate ln(cos(x)).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 13:

Find the derivative of ln(sec(x)).

• Solution:
  • Using the chain rule, we can differentiate ln(sec(x)).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

24. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 14:

Find the derivative of ln(cosec(x)).

• Solution:
  • Using the chain rule, we can differentiate ln(cosec(x)).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 15:

Find the derivative of ln(tan(x)).

• Solution:
  • Using the chain rule, we can differentiate ln(tan(x)).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

25. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 16:

Find the derivative of ln(1/x^2).

• Solution:
  • Using the chain rule, we can differentiate ln(1/x^2).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 17:

Find the derivative of ln(e^(-2x)).

• Solution:
  • Using the chain rule, we can differentiate ln(e^(-2x)).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

26. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 18:

Find the derivative of ln(|x|).

• Solution:
  • Using the chain rule, we can differentiate ln(|x|).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 19:

Find the derivative of ln(|3x|).

• Solution:
  • Using the chain rule, we can differentiate ln(|3x|).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

27. Derivatives - Derivative of ln(x) (Continued)

• The natural logarithm function, ln(x), is the inverse of the exponential function, e^x.
• The derivative of ln(x) is an important derivative in calculus.
• The formula for the derivative of ln(x) is:
  • 
• Let’s understand this concept with some examples.

Example 20:

Find the derivative of ln(3|x|).

• Solution:
  • Using the chain rule, we can differentiate ln(3|x|).
  • Applying the chain rule, we get:
  • 
  • Simplifying further, we find:
  • 

Example 21:

Find the derivative of ln(e^(-3|x|)).

• Solution:
  • Using the chain rule, we can differentiate ln(e^(-3|x|)).
  • Applying the chain rule, we get:
  • )=\frac{1