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Derivatives of Logarithmic Functions

• Introduction to derivatives of logarithmic functions

  • General formulas:

    • ( f(x) = \log_a(x) )

    • ( f(x) = \ln(x) )

Derivatives of Natural Logarithm

• Using implicit differentiation to prove formulas:

  • ( \frac{d}{dx}(\ln x) = \frac{1}{x} )

  • ( \frac{d}{dx}(\ln |x|) = \frac{1}{x} )

Derivatives of Logarithm with Base 'a'

• ( \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} )

• ( \frac{d}{dx}(\log_a |x|) = \frac{1}{x \ln a} )

Example Problems

• Differentiate the following:

  • a) ( R(w) = 4w - 5 \log_9 w )

  • b) ( y = 3e^x + 10x^3 \ln x )

  • c) ( f(x) = \ln(x^3 + 3x - 4) )

More Example Problems

• Continue differentiating:

  • d) ( g(x) = \ln \left( \frac{x^2 \sin x}{2x + 1} \right)

  • e) ( y = 7x )

  • f) ( y = x^7 )

  • g) ( y = 73 )

Finding Tangent Lines

• Problem: Find the equation of the tangent line to the curve ( h(x) = \log_2(3x + 1) ) at ( x = 1 )

Logarithmic Differentiation

• Useful for functions involving products, quotients, or powers:

  1. Take the natural logarithm of both sides: ( y = f(x) \Rightarrow \ln y = \ln f(x) )

     • Note: if ( f(x) < 0 ), use ( |y| = |f(x)| )

  2. Use properties of logarithms to simplify:

     • Product Property: ( \ln(ab) = \ln a + \ln b )

     • Quotient Property: ( \ln \left( \frac{a}{b} \right) = \ln a - \ln b )

     • Power Property: ( \ln(a^p) = p \cdot \ln a )

  3. Differentiate implicitly with respect to ( x ):

     • ( y \cdot \frac{dy}{dx} = \frac{d}{dx} f(x) )

  4. Solve for ( \frac{dy}{dx} ) by multiplying by ( y )

Additional Example

• Differentiate ( y = (1 - 3x) \cos x ) using logarithmic differentiation.

• Prove that ( \frac{d}{dx}(x^n) = nx^{n-1} ) using logarithmic differentiation.

Derivatives of Logarithmic Functions

Introduction to Derivatives of Logarithmic Functions

Understanding the derivatives of logarithmic functions is crucial in calculus as they form an essential part of various applications in science, engineering, and economics. Logarithmic functions help simplify complex calculations when dealing with exponential growth or decay. The two common types of logarithmic functions are the natural logarithm and logarithm with an arbitrary base.

General Formulas:

• Natural Logarithm:( f(x) = ext{ln}(x) ) where the base is Euler's number, approximately 2.718.

• Logarithm with Base 'a':( f(x) = ext{log}_a(x) )

Derivatives of Natural Logarithm

Using implicit differentiation is a common method for deriving formulas related to logarithmic functions. The derivatives are essential in solving problems involving growth rates and optimization.

• Derivative of the Natural Logarithm:( \frac{d}{dx}( ext{ln } x) = \frac{1}{x} )

• Absolute Value of Natural Logarithm:( \frac{d}{dx}( ext{ln } |x|) = \frac{1}{x} )

Derivatives of Logarithm with Base 'a'

For logarithmic functions of any base, we need to apply the chain rule along with the derivative of the natural logarithm to obtain the derivatives.

• Derivative with Base 'a':( \frac{d}{dx}( ext{log}_a x) = \frac{1}{x \ln a} )

• Absolute Value with Base 'a':( \frac{d}{dx}( ext{log}_a |x|) = \frac{1}{x \ln a} )

Example Problems

Differentiate the following functions:

1. ( R(w) = 4w - 5 \text{log}_9 w )

2. ( y = 3e^{x} + 10x^{3} \text{ln } x )

3. ( f(x) = ext{ln}(x^{3} + 3x - 4) )

More Example Problems

Continue Differentiating:

1. ( g(x) = \text{ln } \left( \frac{x^{2} \sin x}{2x + 1} \right) )

2. ( y = 7x )

3. ( y = x^{7} )

4. ( y = 73 )

Finding Tangent Lines

Problem:

Find the equation of the tangent line to the curve ( h(x) = \text{log}_2(3x + 1) ) at ( x = 1 ).To do this, compute ( h'(1) ) and use the point-slope form of the equation of a line.

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique that simplifies the differentiation of products, quotients, or powers. This method is particularly useful when faced with complex functions.

1. Begin by taking the natural logarithm of both sides: ( y = f(x) \Rightarrow \text{ln } y = \text{ln } f(x) )

2. If ( f(x) < 0 ), apply the absolute value: ( |y| = |f(x)| )

3. Utilize properties of logarithms for simplification:

   • Product Property: ( \text{ln}(ab) = \text{ln } a + \text{ln } b )

   • Quotient Property: ( \text{ln} \left( \frac{a}{b} \right) = \text{ln } a - \text{ln } b )

   • Power Property: ( \text{ln}(a^{p}) = p \cdot \text{ln } a )

4. Differentiate implicitly with respect to ( x ): ( y \cdot \frac{dy}{dx} = \frac{d}{dx} f(x) )

5. Isolate ( \frac{dy}{dx} ) by multiplying by ( y ).

Additional Example

Differentiate using Logarithmic Differentiation:

Differentiate ( y = (1 - 3x) \cos x ) to find the derivative.Also, prove that ( \frac{d}{dx}(x^{n}) = nx^{n-1} ) using logarithmic differentiation.

Understanding these techniques and formulas will enhance your ability to tackle advanced calculus problems involving logarithmic functions.