Math Problem Statement
differentiate y=ln(x^2 sin x / (1+x)^1/2 )
Solution
We are asked to differentiate the function:
Step-by-Step Solution:
Step 1: Apply the logarithmic properties
Using the properties of logarithms, we can simplify the expression inside the logarithm:
= \ln(x^2) + \ln(\sin x) - \ln(\sqrt{1+x})$$ Simplifying further: $$y = 2\ln(x) + \ln(\sin x) - \frac{1}{2} \ln(1+x)$$ #### Step 2: Differentiate term-by-term Now, we differentiate each term individually with respect to $$x$$. 1. $$\frac{d}{dx} [2\ln(x)] = \frac{2}{x}$$ 2. $$\frac{d}{dx} [\ln(\sin x)] = \frac{1}{\sin x} \cdot \cos x = \cot x$$ 3. $$\frac{d}{dx} \left[ -\frac{1}{2} \ln(1+x) \right] = -\frac{1}{2} \cdot \frac{1}{1+x} = -\frac{1}{2(1+x)}$$ #### Step 3: Combine the results Now, we combine all the differentiated terms: $$\frac{dy}{dx} = \frac{2}{x} + \cot x - \frac{1}{2(1+x)}$$ Thus, the derivative of $$y$$ is: $$\frac{dy}{dx} = \frac{2}{x} + \cot x - \frac{1}{2(1+x)}$$ Would you like further details or have any questions? Here are 5 related questions to explore: 1. How do you differentiate more complex logarithmic functions? 2. What are the rules for differentiating trigonometric functions? 3. How can the chain rule be applied to logarithmic derivatives? 4. What is the significance of logarithmic differentiation? 5. How do you handle differentiation of products and quotients involving logarithms? **Tip:** Always look for ways to simplify complex functions using logarithmic properties before differentiating!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Differentiation
Logarithmic Differentiation
Trigonometric Functions
Formulas
d/dx[ln(u)] = (1/u) * du/dx
Product Rule: d/dx[u * v] = u'v + uv'
Quotient Rule: d/dx[u / v] = (v*u' - u*v') / v^2
Theorems
Chain Rule
Product Rule
Quotient Rule
Suitable Grade Level
College Level (Calculus)