Differentiate each term using the power rule (d)/(dx)(ax^n) = anx^n-1 and the rule for constants (d)/(dx)(c) = 0.

Q21. Differentiate the following expressions a) Step 1: Differentiate each term using the power rule (d)/(dx)(ax^n) = anx^n-1 and the rule for constants (d)/(dx)(c) = 0. (d)/(dx)(5x^2 + 1) = (d)/(dx)(5x^2) + (d)/(dx)(1) Step 2: Apply the differentiation rules. (d)/(dx)(5x^2) = 5 · 2x^2-1 = 10x (d)/(dx)(1) = 0 Step 3: Combine the results. 10x + 0 = 10x The derivative is: 10x b) Step 1: Use the chain rule for differentiation. Let y = (u) where u = 5x. The chain rule states (dy)/(dx) = (dy)/(du) · (du)/(dx). Step 2: Differentiate y with respect to u. (dy)/(du) = (d)/(du)((u)) = (1)/(u) Step 3: Differentiate u with respect to x. (du)/(dx) = (d)/(dx)(5x) = 5 Step 4: Apply the chain rule by substituting u=5x. (dy)/(dx) = (1)/(u) · 5 = (1)/(5x) · 5 Step 5: Simplify the expression. (5)/(5x) = (1)/(x) The derivative is: (1)/(x) Q22. Compute _r=1^50 (r+2) Step 1: Split the summation into two separate sums. _r=1^50 (r+2) = _r=1^50 r + _r=1^50 2 Step 2: Evaluate the first sum, _r=1^50 r, which is the sum of the first 50 natural numbers. Use the formula _i=1^n i = (n(n+1))/(2). _r=1^50 r = (50(50+1))/(2) = (50 × 51)/(2) = 25 × 51 = 1275 Step 3: Evaluate the second sum, _r=1^50 2, which is summing the constant 2, 50 times. _r=1^50 2 = 50 × 2 = 100 Step 4: Add the results from Step 2 and Step 3. 1275 + 100 = 1375 The computed sum is: 1375