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