How to use logarithmic differentiation?
Understand the Problem
The question is asking for guidance on the method of logarithmic differentiation, which is a technique in calculus used to simplify the differentiation of complex functions.
Answer
Using logarithmic differentiation, the derivative is given by \( \frac{dy}{dx} = f(x) \cdot \text{(right side derivative)} \).
Answer for screen readers
The derivative using logarithmic differentiation is given by: $$ \frac{dy}{dx} = f(x) \cdot \text{(right side derivative)} $$
Steps to Solve
- Introduce the Function
Start with the function you want to differentiate. For example, let ( y = f(x) ).
- Take the Natural Logarithm of Both Sides
Apply the natural logarithm to both sides of the equation: $$ \ln(y) = \ln(f(x)) $$
- Apply Logarithmic Properties
Use properties of logarithms to simplify the right-hand side. If ( f(x) ) is a product or a quotient, you can separate the terms:
- For a product: ( \ln(ab) = \ln(a) + \ln(b) )
- For a quotient: ( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) )
- Differentiate Both Sides with Respect to x
Differentiate both sides of the equation. Remember to use the chain rule on the left side: $$ \frac{d}{dx} [\ln(y)] = \frac{1}{y} \frac{dy}{dx} $$ Now differentiate the right side using the rules established in step 3.
- Solve for ( \frac{dy}{dx} )
Rearrange the equation to isolate ( \frac{dy}{dx} ): $$ \frac{dy}{dx} = y \cdot \text{(right side derivative)} $$
- Substitute Back for y
Finally, replace ( y ) back with the original function ( f(x) ) to express the derivative in terms of the original function: $$ \frac{dy}{dx} = f(x) \cdot \text{(right side derivative)} $$
The derivative using logarithmic differentiation is given by: $$ \frac{dy}{dx} = f(x) \cdot \text{(right side derivative)} $$
More Information
Logarithmic differentiation is especially useful for functions where the dependent variable ( y ) is a product or quotient of other functions, as it simplifies the differentiation process.
Tips
- Forgetting to apply the chain rule when differentiating ( \ln(y) ). Always account for ( \frac{dy}{dx} ).
- Neglecting to simplify the logarithm before differentiating, leading to more complicated derivatives.
- Failing to substitute back for ( y ) at the end, which results in an incomplete expression.
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