This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the differentiations for each function: 21. Differentiate y = x Step 1: Apply the differentiation rule (d)/(dx)( x) = (1)/(x). (dy)/(dx) = (1)/(x) The derivative is (1)/(x). 22. Differentiate y = 5 x Step 1: Apply the constant multiple rule (d)/(dx)(c · f(x)) = c · (d)/(dx)(f(x)). (dy)/(dx) = 5 (d)/(dx)( x) Step 2: Differentiate x. (dy)/(dx) = 5 · (1)/(x) = (5)/(x) The derivative is (5)/(x). 23. Differentiate y = x + 7 Step 1: Apply the sum rule (d)/(dx)(f(x) + g(x)) = (d)/(dx)(f(x)) + (d)/(dx)(g(x)) and the rule for a constant (d)/(dx)(c) = 0. (dy)/(dx) = (d)/(dx)( x) + (d)/(dx)(7) Step 2: Differentiate each term. (dy)/(dx) = (1)/(x) + 0 = (1)/(x) The derivative is (1)/(x). 24. Differentiate y = 3 x - 4 Step 1: Apply the difference rule and constant rules. (dy)/(dx) = (d)/(dx)(3 x) - (d)/(dx)(4) Step 2: Differentiate each term. (dy)/(dx) = 3 · (1)/(x) - 0 = (3)/(x) The derivative is (3)/(x). 25. Differentiate y = 8 x + 5 Step 1: Apply the sum rule and constant rules. (dy)/(dx) = (d)/(dx)(8 x) + (d)/(dx)(5) Step 2: Differentiate each term. (dy)/(dx) = 8 · (1)/(x) + 0 = (8)/(x) The derivative is (8)/(x). 26. Differentiate y = 6 x - 9 Step 1: Apply the difference rule and constant rules. (dy)/(dx) = (d)/(dx)(6 x) - (d)/(dx)(9) Step 2: Differentiate each term. (dy)/(dx) = 6 · (1)/(x) - 0 = (6)/(x) The derivative is (6)/(x). 27. Differentiate y = 10 x + 3 Step 1: Apply the sum rule and constant rules. (dy)/(dx) = (d)/(dx)(10 x) + (d)/(dx)(3) Step 2: Differentiate each term. (dy)/(dx) = 10 · (1)/(x) + 0 = (10)/(x) The derivative is (10)/(x). 28. Differentiate y = 4 x + 2 x Step 1: Simplify the expression. y = (4+2) x = 6 x Step 2: Differentiate the simplified expression. (dy)/(dx) = (d)/(dx)(6 x) = 6 (d)/(dx)( x) Step 3: Differentiate x. (dy)/(dx) = 6 · (1)/(x) = (6)/(x) The derivative is (6)/(x). 29. Differentiate y = 9 x - 3 x Step 1: Simplify the expression. y = (9-3) x = 6 x Step 2: Differentiate the simplified expression. (dy)/(dx) = (d)/(dx)(6 x) = 6 (d)/(dx)( x) Step 3: Differentiate x. (dy)/(dx) = 6 · (1)/(x) = (6)/(x) The derivative is (6)/(x). 30. Differentiate y = 7 x + 5 Step 1: Apply the sum rule and constant rules. (dy)/(dx) = (d)/(dx)(7 x) + (d)/(dx)(5) Step 2: Differentiate each term. (dy)/(dx) = 7 · (1)/(x) + 0 = (7)/(x) The derivative is (7)/(x). Send me the next one 📸