Apply the differentiation rule (d)/(dx)( x) = (1)/(x).

Here are the differentiations for each function: 21. Differentiate $y = \ln x$ Step 1: Apply the differentiation rule $\frac{d}{dx}(\ln x) = \frac{1}{x}$. $$ \frac{dy}{dx} = \frac{1}{x} $$ The derivative is $\boxed{\frac{1}{x}}$. 22. Differentiate $y = 5 \ln x$ Step 1: Apply the constant multiple rule $\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))$. $$ \frac{dy}{dx} = 5 \frac{d}{dx}(\ln x) $$ Step 2: Differentiate $\ln x$. $$ \frac{dy}{dx} = 5 \cdot \frac{1}{x} = \frac{5}{x} $$ The derivative is $\boxed{\frac{5}{x}}$. 23. Differentiate $y = \ln x + 7$ Step 1: Apply the sum rule $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$ and the rule for a constant $\frac{d}{dx}(c) = 0$. $$ \frac{dy}{dx} = \frac{d}{dx}(\ln x) + \frac{d}{dx}(7) $$ Step 2: Differentiate each term. $$ \frac{dy}{dx} = \frac{1}{x} + 0 = \frac{1}{x} $$ The derivative is $\boxed{\frac{1}{x}}$. 24. Differentiate $y = 3 \ln x - 4$ Step 1: Apply the difference rule and constant rules. $$ \frac{dy}{dx} = \frac{d}{dx}(3 \ln x) - \frac{d}{dx}(4) $$ Step 2: Differentiate each term. $$ \frac{dy}{dx} = 3 \cdot \frac{1}{x} - 0 = \frac{3}{x} $$ The derivative is $\boxed{\frac{3}{x}}$. 25. Differentiate $y = 8 \ln x + 5$ Step 1: Apply the sum rule and constant rules. $$ \frac{dy}{dx} = \frac{d}{dx}(8 \ln x) + \frac{d}{dx}(5) $$ Step 2: Differentiate each term. $$ \frac{dy}{dx} = 8 \cdot \frac{1}{x} + 0 = \frac{8}{x} $$ The derivative is $\boxed{\frac{8}{x}}$. 26. Differentiate $y = 6 \ln x - 9$ Step 1: Apply the difference rule and constant rules. $$ \frac{dy}{dx} = \frac{d}{dx}(6 \ln x) - \frac{d}{dx}(9) $$ Step 2: Differentiate each term. $$ \frac{dy}{dx} = 6 \cdot \frac{1}{x} - 0 = \frac{6}{x} $$ The derivative is $\boxed{\frac{6}{x}}$. 27. Differentiate $y = 10 \ln x + 3$ Step 1: Apply the sum rule and constant rules. $$ \frac{dy}{dx} = \frac{d}{dx}(10 \ln x) + \frac{d}{dx}(3) $$ Step 2: Differentiate each term. $$ \frac{dy}{dx} = 10 \cdot \frac{1}{x} + 0 = \frac{10}{x} $$ The derivative is $\boxed{\frac{10}{x}}$. 28. Differentiate $y = 4 \ln x + 2 \ln x$ Step 1: Simplify the expression. $$ y = (4+2) \ln x = 6 \ln x $$ Step 2: Differentiate the simplified expression. $$ \frac{dy}{dx} = \frac{d}{dx}(6 \ln x) = 6 \frac{d}{dx}(\ln x) $$ Step 3: Differentiate $\ln x$. $$ \frac{dy}{dx} = 6 \cdot \frac{1}{x} = \frac{6}{x} $$ The derivative is $\boxed{\frac{6}{x}}$. 29. Differentiate $y = 9 \ln x - 3 \ln x$ Step 1: Simplify the expression. $$ y = (9-3) \ln x = 6 \ln x $$ Step 2: Differentiate the simplified expression. $$ \frac{dy}{dx} = \frac{d}{dx}(6 \ln x) = 6 \frac{d}{dx}(\ln x) $$ Step 3: Differentiate $\ln x$. $$ \frac{dy}{dx} = 6 \cdot \frac{1}{x} = \frac{6}{x} $$ The derivative is $\boxed{\frac{6}{x}}$. 30. Differentiate $y = 7 \ln x + 5$ Step 1: Apply the sum rule and constant rules. $$ \frac{dy}{dx} = \frac{d}{dx}(7 \ln x) + \frac{d}{dx}(5) $$ Step 2: Differentiate each term. $$ \frac{dy}{dx} = 7 \cdot \frac{1}{x} + 0 = \frac{7}{x} $$ The derivative is $\boxed{\frac{7}{x}}$. Send me the next one 📸