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Home » Find the derivative of \( y = \ln(x^2 + 1) \) at \( x = 1 \). - 500apps

Find the derivative of \( y = \ln(x^2 + 1) \) at \( x = 1 \). - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 Question: What is the greatest common factor of 68 and 102?

📌 Solution: We use the prime factorization method. \(68 = 2^2 \times 17\) and \(102 = 2 \times 3 \times 17\). The common prime factors are 2 and 17. Taking the lowest powers, we get \(2^1 \times 17^1 = 34\). Thus, the greatest common factor is \(\boxed{34}\).

📌 Question: How many whole numbers lie in the interval between \(\frac{5}{2}\) and \(3\pi\)?

📌 Solution: First, evaluate the boundary values. \(\frac{5}{2} = 2.5\). Since \(\pi \approx 3.1416\), we have \(3\pi \approx 9.4248\). We seek whole numbers \(n\) such that \(2.5 < n < 9.4248\). These are \(3, 4, 5, 6, 7, 8, 9\). Counting them, there are 7 whole numbers. Therefore, the answer is \(\boxed{7}\).

📌 Question: The average of \(3v+4\), \(5v-2\), and \(4v+7\) is what expression in terms of \(v\)?

📌 Solution: To find the average, sum the three expressions and divide by 3. The sum is \((3v + 4) + (5v - 2) + (4v + 7) = 3v + 5v + 4v + 4 - 2 + 7 = 12v + 9\). Dividing by 3, we get \(\frac{12v + 9}{3} = 4v + 3\). Thus, the average is \(\boxed{4v + 3}\).1. اشترت شركة مقاعد مكتبية وطاولات. كل مكتب يكلف 120 دولارًا، وكل طاولة تكلف 180 دولارًا. إذا أنفقت الشركة 7,800 دولارًا في المجموع واشترت 30 قطعة في المجموع، فكم عدد المقاعد التي اشتروها؟

📌 لتكن \( x \) عدد المكاتب و \( y \) عدد الطاولات.

📌 \( x + y = 30 \)

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