Advanced Differentiation Techniques

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to differentiate functions where both the base and the exponent are functions of x, or when the function is a product or quotient of several functions. This method involves taking the natural logarithm of both sides of the equation and then differentiating implicitly.

• Key Steps:

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

  2. Use logarithm properties to simplify the expression.

  3. Differentiating both sides with respect to $x$ using implicit differentiation.

  4. Solve for $y'$.

• Useful for: Functions of the form $y = [f(x)]^{g(x)}$, products, and quotients of complicated functions.

Example 1: Differentiating $y = \log_a(e^{-x} \cos(n x))$

• Step 1: Rewrite using the change of base formula: $\log_a(u) = \frac{\ln u}{\ln a}$.

• Step 2: Differentiate using the chain rule: $y' = \frac{1}{\ln a} \cdot \frac{1}{e^{-x} \cos(n x)} \cdot \frac{d}{dx}(e^{-x} \cos(n x))$

• Step 3: Apply the product rule to $e^{-x} \cos(n x)$: $\frac{d}{dx}(e^{-x} \cos(n x)) = -e^{-x} \cos(n x) + e^{-x}(-\sin(n x) \cdot n)$

• Final Answer: $y' = \frac{1}{\ln a} \cdot \frac{-e^{-x} \cos(n x) + e^{-x}(-n \sin(n x))}{e^{-x} \cos(n x)}$

Example 2: Differentiating $y = \frac{\ln x}{x^2}$

• Step 1: Use the quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

• Let $u = \ln x$, $v = x^2$

• Compute derivatives: $u' = \frac{1}{x}$, $v' = 2x$

• Apply the quotient rule: $y' = \frac{\frac{1}{x} \cdot x^2 - \ln x \cdot 2x}{x^4} = \frac{x - 2x \ln x}{x^3}$

• Second Derivative: Apply the quotient rule again to $y'$. $y'' = \frac{(x^3 \cdot (1 - 2 \ln x))' - (x - 2x \ln x) \cdot (x^3)'}{(x^3)^2}$ Additional info: The full expansion involves the product and chain rules.

Implicit Differentiation and Logarithmic Properties

Implicit differentiation is used when it is difficult or impossible to solve for $y$ explicitly in terms of $x$. Logarithmic properties can simplify differentiation, especially for products, quotients, or powers.

Example 3: $\ln y = \ln\left(\frac{e^{-x} \cos^2 x}{x^2 + x + 1}\right)$

• Step 1: Use logarithm properties: $\ln y = \ln(e^{-x} \cos^2 x) - \ln(x^2 + x + 1)$ $= -x + 2 \ln(\cos x) - \ln(x^2 + x + 1)$

• Step 2: Differentiate both sides: $\frac{1}{y} y' = -1 + 2 \tan x - \frac{2x + 1}{x^2 + x + 1}$

• Step 3: Solve for $y'$: $y' = y \left(-1 + 2 \tan x - \frac{2x + 1}{x^2 + x + 1}\right)$ Substitute $y$ back: $y' = \frac{e^{-x} \cos^2 x}{x^2 + x + 1} \left(-1 + 2 \tan x - \frac{2x + 1}{x^2 + x + 1}\right)$

Differentiation of Exponential and Logarithmic Functions

When differentiating functions of the form $y = [f(x)]^{g(x)}$, logarithmic differentiation is especially useful.

Example 4: $y = (\sin x)^{\ln x}$

• Step 1: Take the natural logarithm of both sides: $\ln y = \ln((\sin x)^{\ln x}) = \ln x \cdot \ln(\sin x)$

• Step 2: Differentiate both sides: $\frac{1}{y} y' = \frac{1}{x} \ln(\sin x) + \ln x \cdot \cot x$

• Step 3: Solve for $y'$: $y' = (\sin x)^{\ln x} \left( \frac{1}{x} \ln(\sin x) + \ln x \cot x \right)$

Summary Table: Differentiation Techniques Used

Key Point: Logarithmic differentiation simplifies the process of differentiating complex functions, especially those involving products, quotients, or variable exponents.