a) $f(x) = x \cdot b\ln x$ (Çarpım türevi):$$f'(x) = (1) \cdot \ln x + x \cdot \left(\frac{1}{x}\right) = \mathbf{\ln x + 1}$$

b) $f(x) = \frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2}$:$$f'(x) = \frac{4x^3}{4} + \frac{3x^2}{3} + \frac{2x}{2} = \mathbf{x^3 + x^2 + x}$$

c) $f(x) = \frac{\sqrt{x + 2}}{x + 1}$ (Bölüm türevi):$$f'(x) = \frac{\left(\frac{1}{2\sqrt{x+2}}\right)(x + 1) - \sqrt{x + 2}(1)}{(x + 1)^2}$$d) $f(x) = \sin(5x - 2)$ (Zincir kuralı):$$f'(x) = \cos(5x - 2) \cdot (5) = \mathbf{5\cos(5x - 2)}$$