Using Logarithmic Differentiation to Find the Derivative of Complicated Math Functions

Logarithmic Differentiation

Logarithmic differentiation is a technique used to find the derivative of a function, especially when the function is complicated and involves multiple terms, products, or powers

Logarithmic differentiation is a technique used to find the derivative of a function, especially when the function is complicated and involves multiple terms, products, or powers. It involves taking the natural logarithm of both sides of an equation and then differentiating implicitly.

To use logarithmic differentiation, follow these steps:

1. Take the natural logarithm (ln) of both sides of the equation.
2. Apply the properties of logarithms to simplify the equation if possible.
3. Differentiate both sides of the equation implicitly with respect to the variable of interest.
4. Solve the resulting equation for the derivative or the variable being differentiated.

Let’s look at an example to illustrate the use of logarithmic differentiation:

Example:
Find the derivative of the function f(x) = (x^2 + 1)^3 * e^(2x).

Step 1: Take the natural logarithm of both sides:
ln(f(x)) = ln((x^2 + 1)^3 * e^(2x))

Step 2: Use the properties of logarithms to simplify the equation:
ln(f(x)) = 3ln(x^2 + 1) + 2x

Step 3: Differentiate both sides of the equation implicitly with respect to x:
(d/dx) ln(f(x)) = (d/dx) (3ln(x^2 + 1) + 2x)

Using the chain rule on the left side and the product rule on the right side, we get:
(1/f(x))* (d/dx) f(x) = 3(2x)/(x^2 + 1) + 2

Step 4: Solve for the derivative ((d/dx) f(x)):
(d/dx) f(x) = f(x) * (3(2x)/(x^2 + 1) + 2)

Now, substituting f(x) back into the equation gives the final result:
(d/dx) f(x) = (x^2 + 1)^3 * e^(2x) * (3(2x)/(x^2 + 1) + 2)

Logarithmic differentiation is an effective method to find the derivative of a complicated function. It is particularly useful when dealing with expressions involving powers, products, or complicated functions where other differentiation techniques may be cumbersome or impractical.

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