Logarithm Derivative Formula

Finding the Derivative of a Logarithmic Function: Logarithmic Differentiation

Logarithmic Differentiation Explained

Logarithmic differentiation is a technique used in calculus to find the derivative of logarithmic functions. It involves taking the natural logarithm of both sides of the function and then differentiating the resulting equation.

Step-by-Step Process:

1. Take the natural logarithm of both sides of the function.
2. Use the logarithmic property

   ln(a * b) = ln(a) + ln(b)

   to simplify the expression under the logarithm.
3. Differentiate both sides of the simplified equation.
4. Use the chain rule to differentiate the logarithmic term.
5. Solve for the derivative of the original function.

Example: Finding the Derivative of f(x) = ln(8x)

**Step 1: Take the natural logarithm of both sides:**

ln(f(x)) = ln(ln(8x))

**Step 2: Simplify the expression under the logarithm:**

ln(f(x)) = ln(ln(8)) + ln(x)

**Step 3: Differentiate both sides:**

d/dx [ln(f(x))] = d/dx [ln(ln(8)) + ln(x)]

**Step 4: Use the chain rule to differentiate the logarithmic term:**

1/f(x) * df/dx = 0 + 1/x

**Step 5: Solve for df/dx:**

df/dx = f(x) / x

**Therefore, the derivative of f(x) = ln(8x) is: f'(x) = 8/x.**

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