91Ó°ÊÓ

The constant multiple rule is a fundamental concept in differentiation. It states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. In mathematical terms, if you have a function given by \ c \times f(x) \ where \ c \ is a constant, then the derivative is: \[ \frac{d}{dx} \big[c \times f(x)\big] = c \times \frac{d}{dx}[f(x)] \] This rule simplifies complex differentiation tasks, making it easier to handle functions with constants. In our exercise, applying the constant multiple rule helps to separately differentiate each part of the function \ \frac{3 \sin x}{2} \text{ and } -\frac{5 \cos x}{8} \.

The derivative of the sine function is key to solving trigonometric differentiation problems. If you have the function \ \sin x \, its derivative is given by: \[ \frac{d}{dx} [\sin x] = \cos x \] This means that the rate of change of the sine function at any point is equal to the cosine of that point. In our problem, when differentiating \ \frac{3 \sin x}{2} \, we apply the constant multiple rule along with the derivative of \ \sin x \ to get \[ \frac{d}{dx} \bigg[ \frac{3 \sin x}{2} \bigg] = \frac{3}{2} \cos x \] This simplifies the differentiation process significantly.

Just like the sine function, the cosine function has a specific derivative formula. If you have the function \ \cos x \, its derivative is given by: \[ \frac{d}{dx} [\cos x] = -\sin x \] This means that the rate of change of the cosine function at any point is the negative sine of that point. For our function, \ -\frac{5 \cos x}{8} \, we first notice the constant of \ -\frac{5}{8} \ and apply the constant multiple rule. Then, we use the derivative of cosine to get: \[ \frac{d}{dx} \bigg[ - \frac{5 \cos x}{8} \bigg] = - \frac{5}{8} \left( - \sin x \right) = \frac{5}{8} \sin x \] Combining all these, you get the final derivative of the given function as \[ f'(x) = \frac{3}{2} \cos x + \frac{5}{8} \sin x \]