Notes on Derivatives of Logarithmic Functions

Let's start exploring the derivatives of logarithmic functions together! We’ll work through the concepts step-by-step so you can understand everything clearly.

Basic Derivatives

  1. Understanding Logarithms:
    A logarithm tells us the power to which a number must be raised to obtain another number. For example, the logarithm base 10 of 100 is 2, because 10² = 100.

  2. Derivative of a logarithm with base a:
    The derivative of a logarithm with any base (let's call it 'a') is given by this formula:
    ddxlog⁡ax=1xln⁡a\frac{d}{dx} \log_a x = \frac{1}{x \ln a}
    Here, (\ln a) is the natural logarithm of 'a'. Don't worry too much about the ln right now; it's just a special number that we will use in derivatives.

  3. Derivative of the natural logarithm:
    The natural logarithm is denoted as (\ln x) and it has its own simple derivative:
    ddxln⁡x=1x\frac{d}{dx} \ln x = \frac{1}{x}
    This means that if we have a function involving natural logarithms, we can easily find its rate of change (or derivative).

Applying Derivatives to Examples:
Example 1: Differentiate ( y = \ln(3x^2 + 4x) )

  • First, we recognize that this isn't a straightforward natural logarithm; it's a logarithm of a more complicated function (the expression inside the ln).

  • To differentiate this, we'll use the chain rule, which is like peeling an onion layer by layer.

    • Here, let’s define:

      • ( g(x) = 3x^2 + 4x ) (this is the inside function)

      • The derivative of ( g(x) ) is:
        g′(x)=ddx(3x2+4x)=6x+4g'(x) = \frac{d}{dx}(3x^2 + 4x) = 6x + 4

    • Now we plug it into our chain rule formula:
      dydx=g′(x)g(x)\frac{dy}{dx} = \frac{g'(x)}{g(x)}

    • Thus, we have:
      dydx=6x+43x2+4x\frac{dy}{dx} = \frac{6x + 4}{3x^2 + 4x}

Example 2: Differentiate ( y = \ln(\cos x) )

  • Similar to before, we apply the chain rule. Here, the inside function is ( \cos x ).

  • The derivative of ( \cos x ) is ( -\sin x ), so we have:
    dydx=−sin⁡xcos⁡x=−tan⁡x\frac{dy}{dx} = \frac{- \sin x}{\cos x} = -\tan x

Example 3: Differentiate ( \frac{\ln(\sqrt{x-2})}{x+1} )

  • Here, we apply both the quotient rule (since we have a fraction) and the chain rule for the logarithm.

  • Let’s define:

    • ( u = \ln(\sqrt{x-2}) )

    • ( v = x + 1 )

  • According to the quotient rule:
    dydx=vdudx−udvdxv2\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}

  • You would differentiate each piece like we did with earlier examples—just keep applying the rules we've learned!

Example 4: Find ( f'(x) ) if ( f(x) = \ln |x| )

  • The derivative here is given by a simple formula:
    ddxln⁡∣x∣=1x\frac{d}{dx} \ln|x| = \frac{1}{x}

  • This derivative is especially important in further calculus studies, letting you compute rates of change efficiently.

Logarithmic Differentiation

  • Purpose: Logarithmic differentiation is a technique that simplifies differentiation when dealing with complicated functions.

  • Let's say we want to differentiate ( y = x^{3/4}\sqrt{x^2 + 1}(3x + 2)^5 ).

  • Here’s how we’ll do it step-by-step:

    1. Take the natural logarithm of both sides:
      ln⁡y=34ln⁡x+12ln⁡(x2+1)+5ln⁡(3x+2)\ln y = \frac{3}{4} \ln x + \frac{1}{2} \ln(x^2 + 1) + 5 \ln(3x + 2)

    2. Now, differentiate implicitly with respect to x.

    3. Finally, solve for ( \frac{dy}{dx} ) by rearranging our equation.

Conclusion

  • Steps of Logarithmic Differentiation:

  1. Take natural logarithms of both sides and apply logarithm laws to simplify.

  2. Differentiate implicitly with respect to ( x ).

  3. Solve for the derivative ( \frac{dy}{dx} ).

Proof of the Power Rule

  • Let’s also find ( \frac{dy}{dx} ) if ( y = x\sqrt{x} ).

  • We can apply the product rule (which you use when multiplying functions together).

  • Rewrite it as ( y = x^{3/2} ).

  • Now differentiate it:
    dydx=32x1/2\frac{dy}{dx} = \frac{3}{2} x^{1/2}

  • If you need to confirm your result, you can check via logarithmic